Eigenvalue distributions of high-dimensional matrix processes driven by fractional Brownian motion

Jian Songlabel=e1][email protected] [ Jianfeng Yao label=e2][email protected] [ Wangjun Yuanlabel=e3][email protected] [ Shandong University and The University of Hong Kong Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, Shandong, 266237, China and School of Mathematics, Shandong University, Jinan, Shandong, 250100, China
Department of Statistics and Actuarial Science, The University of Hong Kong
Department of Mathematics, The University of Hong Kong

Abstract

In this article, we study high-dimensional behavior of empirical spectral distributions $\{L_{N}(t),t\in[0,T]\}$ for a class of $N\times N$ symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H\in(1/2,1)$ . For Wigner-type matrices, we obtain almost sure relative compactness of $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ in $C([0,T],\mathbf{P}(\mathbb{R}))$ following the approach in [Anderson2010]; for Wishart-type matrices, we obtain tightness of $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ on $C([0,T],\mathbf{P}(\mathbb{R}))$ by tightness criterions provided in Appendix B. The limit of $\{L_{N}(t),t\in[0,T]\}$ as $N\to\infty$ is also characterised.

60H15, 60F05,

matrix-valued process,

fractional Brownian motion,

empirical spectral distribution,

measure-valued process,

tightness criterion,

keywords:

[class=AMS]

keywords:

\startlocaldefs\endlocaldefs

and and

1 Introduction

There has been increasing research activity on matrix-valued stochastic differential equations (SDEs) in recent years. A prominent example is the class of generalized Wishart processes introduced in [Graczyk2013]. This class generalizes classical examples of symmetric Brownian motion [Dyson62], the Wishart process [Bru91] and the symmetric matrix-valued process whose entries are independent Ornstein-Uhlenbeck processes [Chan1992]. Recent works on generalized Wishart processes include [Song20, Song19] and [Graczyk2014, Graczyk2019]. A common feature of these processes is that they are all driven by independent Brownian motions.

In contrast, the study of SDE matrices driven by fractional Brownian motions (fBm) has a shorter history and less literature. To our best knowledge, the first paper is [Nualart20144266], where the symmetric fractional Brownian matrix was studied. The SDE for the associated eigenvalues and conditions for non-collision of the eigenvalues were obtained when the Hurst parameter $H>1/2$ by using fractional calculus and Malliavin calculus. The convergence in distribution of the empirical spectral measure of a scaled symmetric fractional Brownian matrix was established in [Pardo2016] by using Malliavin calculus and a tightness argument. These results were generalized to centered Gaussian process in [Jaramillo2019]. Besides, [Pardo2017] obtained SDEs for the eigenvalues, conditions for non-collision of eigenvalues and the convergence in law of the empirical eigenvalue measure processes for a scaled fractional Wishart matrix (the product of an independent fractional Brownian matrix and its transpose). In the present paper, by adopting a different approach, we obtain stronger results on the convergence of empirical measures of eigenvalues for a significantly larger class of matrix-valued processes driven by fBms (see Remark 3.1).

More precisely, consider the following 1-dimensional SDE

\displaystyle dX_{t}=\sigma(X_{t})\circ dB_{t}^{H}+b(X_{t})dt,\quad t\geq 0,

(1.1)

with initial value $X_{0}$ independent of $B^{H}=(B^{H}_{t})_{t\geq 0}$ . Here, $B_{t}^{H}$ is a fractional Brownian motion with Hurst parameter $H\in(1/2,1)$ , and the differential $\circ dB_{t}^{H}$ is in the Stratonovich sense. It has been shown in [Lyons1994] that there is a unique solution to SDE (1.1) if the coefficient functions $\sigma$ and $b$ have bounded derivatives which are Hölder continuous of order greater than $1/H-1$ .

Let $\{X_{ij}(t)\}_{i,j\geq 1}$ be i.i.d. copies of $X_{t}$ and $Y^{N}(t)=\left(Y_{ij}^{N}(t)\right)_{1\leq i,j\leq N}$ be a symmetric $N\times N$ matrix with entries

\displaystyle Y_{ij}^{N}(t)=\begin{cases}\dfrac{1}{\sqrt{N}}X_{ij}(t),&1\leq i<j\leq N,\\ \dfrac{\sqrt{2}}{\sqrt{N}}X_{ii}(t),&1\leq i\leq N.\end{cases}

(1.2)

Let $\lambda_{1}^{N}(t)\leq\cdots\leq\lambda_{N}^{N}(t)$ be the eigenvalues of $Y^{N}(t)$ and

\displaystyle L_{N}(t)(dx)=\dfrac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}^{N}(t)}(dx)

(1.3)

be the empirical distribution of the eigenvalues.

In this paper, we aim to study, as the dimension $N$ tends to infinity, the limiting behavior of the empirical measure-valued processes $\{L_{N}(t),t\in[0,T]\}$ defined by (1.3) and the empirical measure-valued processes arising from other related matrix-valued processes in the space $C([0,T],\mathbf{P}(\mathbb{R}))$ of continuous measure-valued processes, with $\mathbf{P}(\mathbb{R})$ being the space of probability measures equipped with its weak topology. A summary of the contents of the paper is as follows.

In Section 2, we study the solution of SDE (1.1). In [Hu2007], a pathwise upper bound for $|X_{t}|$ on the time interval $[0,T]$ was obtained under some boundedness and smoothness conditions on the coefficient functions. We adapt the techniques used in [Hu2007] to study the increments $|X_{t}-X_{s}|$ for $t,s\in[0,T]$ and obtain the pathwise Hölder continuity for $X_{t}$ . Moreover, under some integrability conditions on the initial value $X_{0}$ , the Hölder norm of $X_{t}$ is shown to be $L^{p}$ -integrable. These results improve the previous results in [nr02] with sharper bounds for the increments of the process.

In Section 3, we prove the almost sure high-dimensional convergence in $C([0,T],\mathbf{P}(\mathbb{R}))$ for the empirical spectral measure-valued process (1.3) of the Wigner-type matrix (1.2). We also obtain the PDE for the Stieltjes transform $G_{t}(z)$ of the limiting measure process $\mu_{t}$ , which turns to be the complex Burgers’ equation up to a change of variable (see Remark 3.2 and Remark 3.3). This generalizes the results of [Pardo2016], where the special case of $X_{t}=B_{t}^{H}$ was studied (note that the convergence obtained therein is in law). A complex analogue is also studied at the end of this section.

In Section 4, we extend the model of Wigner-type matrix to the locally dependent symmetric matrix-valued stochastic process $R^{N}(t)=\left(R^{N}_{ij}(t)\right)_{1\leq i,j\leq N}$ , where each $R_{ij}^{N}(t)$ is a weighted sum of i.i.d. $\{X_{(k,l)}(t)\}$ for $(k,l)$ in a fixed and bounded neighborhood of $(i,j)$ (see (4.1) for the definition). We also establish the almost sure convergence of the empirical spectral measure-valued process in $C([0,T],\mathbf{P}(\mathbb{R}))$ . Moreover, the limiting measure-valued process $\mu_{t}$ is characterized by an equation satisfied by its Stieltjes transform (see Theorem 4.1). It is worth noticing that for the proofs of the almost sure convergence for the empirical spectral measure-valued processes in both Section 3 and Section 4, we follow the strategy used in [Song20] (which was inspired by [Anderson2010]).

In Section 5, we study the high-dimensional limit of the empirical spectral measure-valued processes of the Wishart-type matrix-valued processes given by (5.1) and its complex analogue (5.9). In the special case where $X_{t}=B_{t}^{H}$ , we recover the convergence result in [Pardo2017]. We also obtain the PDE for the Stieltjes transform $G_{t}(z)$ of the limit measure process $\mu_{t}$ (see Remarks 5.4 and 5.6).

For Wishart-type matrix-valued processes, we are not able to obtain the almost sure convergence in $C([0,T],\mathbf{P}(\mathbb{R}))$ for the empirical spectral measure-valued processes as done in Section 3 and Section 4 since the method used there heavily relies on the independence of the upper triangular entries. Instead, we obtain the convergence in law by a tightness criterion for probability measures on $C([0,T],\mathbf{P}(\mathbb{R}))$ (see Theorem B.2, Propositions B.2 and B.3 in Appendix B). The tightness criterion Theorem B.2 might be already well-known in the literature (we refer to, e.g., [Rogers1993, Pardo2016, Pardo2017, Jaramillo2019] for related arguments), however, we could not find a reference stating it explicitly, and we provide it in Appendix B.

Finally, for the reader’s convenience, some preliminaries on random matrices used in the proofs are provided in Appendix A, and some useful lemmas and tightness criterions are provided in Appendix B.

2 Hölder continuity of the solution to the SDE (1.1)

Some notations are in order. The Hölder norm of a Hölder continuous function $f$ of order $\beta$ is

\displaystyle\|f\|_{a,b;\beta}=\sup_{a\leq x<y\leq b}\dfrac{|f(x)-f(y)|}{|x-y|^{\beta}}.

We also use

\displaystyle\|M\|_{F}=\left(\sum_{i,j=1}^{N}|M_{ij}|^{2}\right)^{1/2}.

to denote the Frobenius norm (also known as the Hilbert-Schmidt norm or $2$ -Schatten norm) of a $N\times N$ matrix $M=(M_{ij})_{1\leq i,j\leq N}$ .

2.1 Preliminaries on fractional calculus and fractional Brownian motion

In this subsection, we recall some basic results in fractional calculus. See [skm93] for more details. Let $a,b\in\mathbb{R}$ with $a<b$ and let $\alpha>0.$ The left-sided and right-sided fractional Riemann-Liouville integrals of $f\in L^{1}([a,b])$ of order $\alpha$ are defined for almost all $t\in(a,b)$ by

I_{a+}^{\alpha}f\left(t\right)=\frac{1}{\Gamma\left(\alpha\right)}\int_{a}^{t}\left(t-s\right)^{\alpha-1}f\left(s\right)ds\,,

and

I_{b-}^{\alpha}f\left(t\right)=\frac{\left(-1\right)^{-\alpha}}{\Gamma\left(\alpha\right)}\int_{t}^{b}\left(s-t\right)^{\alpha-1}f\left(s\right)ds,

respectively, where $\left(-1\right)^{-\alpha}=e^{-i\pi\alpha}$ and $\displaystyle\Gamma\left(\alpha\right)=\int_{0}^{\infty}r^{\alpha-1}e^{-r}dr$ is the Euler gamma function.

Let $I_{a+}^{\alpha}(L^{p})$ (resp. $I_{b-}^{\alpha}(L^{p})$ ) be the image of $L^{p}([a,b])$ by the operator $I_{a+}^{\alpha}$ (resp. $I_{b-}^{\alpha}$ ). If $f\in I_{a+}^{\alpha}(L^{p})$ (resp. $f\in I_{b-}^{\alpha}(L^{p})$ ) and $\alpha\in(0,1)$ , then the left-sided and right-sided fractional derivatives are defined as

	$\displaystyle D_{a+}^{\alpha}f\left(t\right)=\frac{1}{\Gamma\left(1-\alpha\right)}\left(\frac{f\left(t\right)}{\left(t-a\right)^{\alpha}}+\alpha\int_{a}^{t}\frac{f\left(t\right)-f\left(s\right)}{\left(t-s\right)^{\alpha+1}}ds\right),$
	$\displaystyle D_{b-}^{\alpha}f\left(t\right)=\frac{\left(-1\right)^{\alpha}}{\Gamma\left(1-\alpha\right)}\left(\frac{f\left(t\right)}{\left(b-t\right)^{\alpha}}+\alpha\int_{t}^{b}\frac{f\left(t\right)-f\left(s\right)}{\left(s-t\right)^{\alpha+1}}ds\right)$		(2.1)

respectively, for almost all $t\in(a,b)$ .

Let $C^{\alpha}([a,b])$ denote the space of $\alpha$ -Hölder continuous functions of order $\alpha$ on the interval $[a,b]$ . When $\alpha p>1$ , then we have $I_{a+}^{\alpha}(L^{p})\subset C^{\alpha-\frac{1}{p}}([a,b])$ . On the other hand, if $\beta>\alpha$ , then $C^{\beta}([a,b])\subset I_{a+}^{\alpha}(L^{p})$ for all $p>1$ .

The following inversion formulas hold:

	$\displaystyle I_{a+}^{\alpha}(I_{a+}^{\beta}f)=I_{a+}^{\alpha+\beta}f,$		$\displaystyle f\in L^{1};$
	$\displaystyle D_{a+}^{\alpha}(I_{a+}^{\alpha}f)=f,$		$\displaystyle f\in L^{1};$
	$\displaystyle I_{a+}^{\alpha}(D_{a+}^{\alpha}f)=f,$		$\displaystyle f\in I_{a+}^{\alpha}(L^{1});$
	$\displaystyle D_{a+}^{\alpha}(D_{a+}^{\beta}f)=D_{a+}^{\alpha+\beta}f,$		$\displaystyle f\in I_{a+}^{\alpha+\beta}(L^{1}),~{}\alpha+\beta\leq 1.$

Similar inversion formulas hold for the operators $I_{b-}^{\alpha}$ and $D_{b-}^{\alpha}$ as well.

We also have the following integration by parts formula.

Proposition 2.1.

If $f\in I_{a+}^{\alpha}(L^{p}),~{}g\in I_{b-}^{\alpha}(L^{q})$ and $\frac{1}{p}+\frac{1}{q}=1$ , we have

\int_{a}^{b}(D_{a+}^{\alpha}f)(s)g(s)ds=\int_{a}^{b}f(s)(D_{b-}^{\alpha}g)(s)ds.

(2.2)

The following proposition indicates the relationship between Young’s integral and Lebesgue integral.

Proposition 2.2.

Suppose that $f\in C^{\lambda}(a,b)$ and $g\in C^{\mu}(a,b)$ with $\lambda+\mu>1$ . Let ${\lambda}>\alpha$ and $\mu>1-\alpha$ . Then the Riemann-Stieltjes integral $\int_{a}^{b}fdg$ exists and it can be expressed as

\int_{a}^{b}fdg=(-1)^{\alpha}\int_{a}^{b}D_{a+}^{\alpha}f\left(t\right)D_{b-}^{1-\alpha}g_{b-}\left(t\right)dt\,,

(2.3)

where $g_{b-}\left(t\right)=g\left(t\right)-g\left(b\right)$ .

It is known that almost all the paths of $B^{H}$ are $(H-\varepsilon)$ -Hölder continuous for $\varepsilon\in(0,H)$ . By the Fernique Theorem, we have the following estimation for its Hölder norm.

Lemma 2.1.

There exists a positive constant $\alpha=\alpha(H,\varepsilon,T)$ depending on $(H,\varepsilon,T)$ , such that

\displaystyle\mathbb{E}\left[e^{\alpha\|B^{H}\|_{0,T;H-\varepsilon}^{2}}\right]<\infty.

2.2 Hölder continuity

In this subsection, for the solution to (1.1), we provide some estimations for its Hölder norm, following the approach developed in [Hu2007, Theorem 2].

Theorem 2.1.

Suppose that the coefficient functions $\sigma$ and $b$ are bounded and have bounded derivatives which are Hölder continuous of order greater than $1/(H-\varepsilon)-1$ for some $\varepsilon\in(0,H-\frac{1}{2})$ . Then there exists a constant $C(H,\varepsilon)$ that depends on $(H,\varepsilon)$ only, such that for all $T>0$ and $0\leq s<t\leq T$ ,

	$\displaystyle\|X_{t}-X_{s}\|$	$\displaystyle\leq C(H,\varepsilon)\\|\sigma\\|_{\infty}\Big{[}\\|B^{H}\\|_{0,T;H-\varepsilon}(t-s)^{H-\varepsilon}\vee\\|B^{H}\\|_{0,T;H-\varepsilon}^{1/(H-\varepsilon)}\\|\sigma^{\prime}\\|_{\infty}^{1/(H-\varepsilon)-1}(t-s)\Big{]}$
		$\displaystyle\quad+2\\|b\\|_{\infty}(t-s).$		(2.4)

Consequently, there exists a random variable $\xi$ with $\mathbb{E}|\xi|^{p}<\infty$ for all $p>1$ such that for $0\leq s<t\leq T$ ,

|X_{t}(\omega)-X_{s}(\omega)|\leq\xi(\omega)|t-s|^{H-\varepsilon},a.s.

Remark 2.1.

Note that the pathwise Hölder continuity of the process $X_{t}$ was established in [nr02, Theorem 2.1] under some Lipschitz conditions and growth conditions on the coefficient functions $b$ and $\sigma$ . They also obtained the $L^{p}$ -integrability of the Hölder norm of the process for $H>\frac{1}{2}+\frac{\gamma}{4}$ where $\gamma\in[0,1]$ is the growth rate of the function $\sigma$ .

An upper bound for $\sup_{t\in[0,T]}|X_{t}|$ was obtained in [Hu2007, Theorem 2], and it turns out that the techniques introduced in [Hu2007] can be used to improve the estimations for the pathwise Hölder norm. This is done in Theorem 2.1 for bounded coefficient functions, and in Theorem 2.2 for unbounded coefficient functions (with linear growth). Compared with the result in [nr02], Theorems 2.1 and 2.2 provide sharper upper bounds for the increments of the processes, which allow to establish the $L^{p}$ -integrability of the Hölder norm for all $H>1/2$ . In contrast, [nr02, Theorem 2.1] obtained the $L^{p}$ -integrability for $H>3/4$ when the coefficient functions have linear growth.

Proof of Theorem 2.1.

Fix $\alpha\in(1-H+\varepsilon,1/2)$ . By Proposition 2.2, for $0\leq s\leq t\leq T$ ,

\displaystyle\left|\int_{s}^{t}\sigma(X_{r})\circ dB_{r}^{H}\right|

\displaystyle\leq\int_{s}^{t}\left|D_{s+}^{\alpha}\sigma(X_{r})D_{t-}^{1-\alpha}(B_{r}^{H}-B_{t}^{H})\right|dr.

(2.5)

By (2.1) and Lemma 2.1, for $r\in[s,t]$ , we have

$\displaystyle\left\|D_{t-}^{1-\alpha}(B_{r}^{H}-B_{t}^{H})\right\|$	$\displaystyle=\dfrac{1}{\Gamma(\alpha)}\left\|\dfrac{B_{r}^{H}-B_{t}^{H}}{(t-r)^{1-\alpha}}+(1-\alpha)\int_{r}^{t}\dfrac{B_{r}^{H}-B_{u}^{H}}{(u-r)^{2-\alpha}}du\right\|$
	$\displaystyle\leq\dfrac{1}{\Gamma(\alpha)}\dfrac{\left\|B_{r}^{H}-B_{t}^{H}\right\|}{\|t-r\|^{1-\alpha}}+\dfrac{1-\alpha}{\Gamma(\alpha)}\int_{r}^{t}\dfrac{\left\|B_{r}^{H}-B_{u}^{H}\right\|}{\|u-r\|^{2-\alpha}}du$
	$\displaystyle\leq\dfrac{\\|B^{H}\\|_{s,t;H-\varepsilon}}{\Gamma(\alpha)}\|t-r\|^{H-\varepsilon-1+\alpha}+\dfrac{(1-\alpha)\\|B^{H}\\|_{s,t;H-\varepsilon}}{\Gamma(\alpha)}\int_{r}^{t}(u-r)^{H-\varepsilon+\alpha-2}du$
	$\displaystyle=\left(\dfrac{1}{\Gamma(\alpha)}+\dfrac{1-\alpha}{(H-\varepsilon+\alpha-1)\Gamma(\alpha)}\right)\\|B^{H}\\|_{s,t;H-\varepsilon}\|t-r\|^{H-\varepsilon-1+\alpha}\,,$	(2.6)

and

$\displaystyle\left\|D_{s+}^{\alpha}\sigma(X_{r})\right\|$	$\displaystyle\leq\dfrac{1}{\Gamma(1-\alpha)}\dfrac{\left\|\sigma(X_{r})\right\|}{\|r-s\|^{\alpha}}+\dfrac{\alpha}{\Gamma(1-\alpha)}\int_{s}^{r}\dfrac{\left\|\sigma(X_{r})-\sigma(X_{u})\right\|}{\|r-u\|^{\alpha+1}}du$
	$\displaystyle\leq\dfrac{\\|\sigma\\|_{\infty}}{\Gamma(1-\alpha)}\|r-s\|^{-\alpha}+\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}}{\Gamma(1-\alpha)}\int_{s}^{r}\dfrac{\left\|X_{r}-X_{u}\right\|}{\|r-u\|^{\alpha+1}}du$
	$\displaystyle\leq\dfrac{\\|\sigma\\|_{\infty}}{\Gamma(1-\alpha)}\|r-s\|^{-\alpha}+\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)}\int_{s}^{r}\dfrac{1}{\|r-u\|^{\alpha+1-(H-\varepsilon)}}du$
	$\displaystyle=\dfrac{\\|\sigma\\|_{\infty}}{\Gamma(1-\alpha)}\|r-s\|^{-\alpha}+\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)(H-\varepsilon-\alpha)}\|r-s\|^{H-\varepsilon-\alpha}\,,$	(2.7)

where the Hölder norm $\|X\|_{s,t;H-\varepsilon}$ is finite a.s. by [nr02, Theorem 2.1].

By (2.5), (2.2) and (2.2), we have

	$\displaystyle\left\|\int_{s}^{t}\sigma(X_{r})\circ dB_{r}^{H}\right\|$
$\displaystyle\leq$	$\displaystyle\int_{s}^{t}\left(\dfrac{\\|\sigma\\|_{\infty}}{\Gamma(1-\alpha)}\|r-s\|^{-\alpha}+\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)(H-\varepsilon-\alpha)}\|r-s\|^{H-\varepsilon-\alpha}\right)$
	$\displaystyle\qquad\quad\left(\dfrac{1}{\Gamma(\alpha)}+\dfrac{1-\alpha}{(H-\varepsilon+\alpha-1)\Gamma(\alpha)}\right)\\|B^{H}\\|_{s,t;H-\varepsilon}\|t-r\|^{H-\varepsilon-1+\alpha}dr$
$\displaystyle=$	$\displaystyle\left(\dfrac{1}{\Gamma(\alpha)}+\dfrac{1-\alpha}{(H-\varepsilon+\alpha-1)\Gamma(\alpha)}\right)\dfrac{\\|\sigma\\|_{\infty}}{\Gamma(1-\alpha)}\\|B^{H}\\|_{s,t;H-\varepsilon}\int_{s}^{t}\|t-r\|^{H-\varepsilon-1+\alpha}\|r-s\|^{-\alpha}dr$
	$\displaystyle+\left(\dfrac{1}{\Gamma(\alpha)}+\dfrac{1-\alpha}{(H-\varepsilon+\alpha-1)\Gamma(\alpha)}\right)\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)(H-\varepsilon-\alpha)}\\|B^{H}\\|_{s,t;H-\varepsilon}\int_{s}^{t}\|t-r\|^{H-\varepsilon-1+\alpha}\|r-s\|^{H-\varepsilon-\alpha}dr$
$\displaystyle=$	$\displaystyle\left(\dfrac{1}{\Gamma(\alpha)}+\dfrac{1-\alpha}{(H-\varepsilon+\alpha-1)\Gamma(\alpha)}\right)\dfrac{\\|\sigma\\|_{\infty}}{\Gamma(1-\alpha)}\\|B^{H}\\|_{s,t;H-\varepsilon}(t-s)^{H-\varepsilon}\beta(H-\varepsilon+\alpha,1-\alpha)$
	$\displaystyle\quad+\left(\dfrac{1}{\Gamma(\alpha)}+\dfrac{1-\alpha}{(H-\varepsilon+\alpha-1)\Gamma(\alpha)}\right)\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)(H-\varepsilon-\alpha)}\\|B^{H}\\|_{s,t;H-\varepsilon}(t-s)^{2H-2\varepsilon}$
	$\displaystyle\quad\qquad\times\beta(H-\varepsilon+\alpha,H-\varepsilon-\alpha+1)$
$\displaystyle\leq$	$\displaystyle C_{1}(H,\varepsilon)\\|B^{H}\\|_{0,T;H-\varepsilon}\Big{[}\\|\sigma\\|_{\infty}(t-s)^{H-\varepsilon}+\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}(t-s)^{2H-2\varepsilon}\Big{]},$	(2.8)

where $\beta(p,q)$ is the Beta function, and $C_{1}(H,\varepsilon)$ is a constant depending on $(H,\varepsilon)$ only.

Hence, by (2.2), for $0\leq s\leq t\leq T$ ,

$\displaystyle\left\|X_{t}-X_{s}\right\|$	$\displaystyle\leq\left\|\int_{s}^{t}\sigma(X_{r})\circ dB_{r}^{H}\right\|+\left\|\int_{s}^{t}b(X_{r})dr\right\|$
	$\displaystyle\leq C_{1}(H,\varepsilon)\\|B^{H}\\|_{0,T;H-\varepsilon}\Big{[}\\|\sigma\\|_{\infty}(t-s)^{H-\varepsilon}+\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}(t-s)^{2H-2\varepsilon}\Big{]}$
	$\displaystyle\quad+\\|b\\|_{\infty}(t-s),$	(2.9)

and therefore,

	$\displaystyle\\|X\\|_{s,t;H-\varepsilon}$	$\displaystyle\leq C_{1}(H,\varepsilon)\\|B^{H}\\|_{0,T;H-\varepsilon}\Big{[}\\|\sigma\\|_{\infty}+\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}(t-s)^{H-\varepsilon}\Big{]}$
		$\displaystyle\quad+\\|b\\|_{\infty}(t-s)^{1-H+\varepsilon}.$		(2.10)

Choose $\Delta$ such that

\displaystyle\Delta^{H-\varepsilon}=\dfrac{1}{2C_{1}(H,\varepsilon)\|B^{H}\|_{0,T;H-\varepsilon}\|\sigma^{\prime}\|_{\infty}}\,,

and if the denominator vanishes, we set $\Delta=\infty$ .

If $\Delta\geq t-s$ , (2.2) yields

\displaystyle\|X\|_{s,t;H-\varepsilon}

\displaystyle\leq C_{1}(H,\varepsilon)\|B^{H}\|_{0,T;H-\varepsilon}\|\sigma\|_{\infty}+\dfrac{1}{2}\|X\|_{s,t;H-\varepsilon}+\|b\|_{\infty}(t-s)^{1-H+\varepsilon},

and thus,

	$\displaystyle\left\|X_{t}-X_{s}\right\|$	$\displaystyle\leq(t-s)^{H-\varepsilon}\\|X\\|_{s,t;H-\varepsilon}$
		$\displaystyle\leq 2C_{1}(H,\varepsilon)\\|B^{H}\\|_{0,T;H-\varepsilon}\\|\sigma\\|_{\infty}(t-s)^{H-\varepsilon}+2\\|b\\|_{\infty}(t-s).$		(2.11)

If $\Delta<t-s$ , we divide the interval $[s,t]$ into $n=[(t-s)/\Delta]+1$ subintervals, whose lengths are smaller than $\Delta$ . Let $s=u_{0}<u_{1}<\cdots<u_{n}=t$ be the endpoints of the subintervals. Then $u_{i}-u_{i-1}\leq\Delta$ for $1\leq i\leq n$ . Hence, by (2.2),

$\displaystyle\left\|X_{t}-X_{s}\right\|$	$\displaystyle\leq\sum_{i=1}^{n}\left\|X_{u_{i}}-X_{u_{i-1}}\right\|$
	$\displaystyle\leq 2C_{1}(H,\varepsilon)\\|B^{H}\\|_{0,T;H-\varepsilon}\\|\sigma\\|_{\infty}\sum_{i=1}^{n}(u_{i}-u_{i-1})^{H-\varepsilon}+2\\|b\\|_{\infty}\sum_{i=1}^{n}(u_{i}-u_{i-1})$
	$\displaystyle\leq 2C_{1}(H,\varepsilon)\\|B^{H}\\|_{0,T;H-\varepsilon}\\|\sigma\\|_{\infty}n\Delta^{H-\varepsilon}+2\\|b\\|_{\infty}(t-s)$
	$\displaystyle\leq 4C_{1}(H,\varepsilon)\\|B^{H}\\|_{0,T;H-\varepsilon}\\|\sigma\\|_{\infty}(t-s)\Delta^{H-\varepsilon-1}+2\\|b\\|_{\infty}(t-s)$
	$\displaystyle\leq 2^{1+1/(H-\varepsilon)}C_{1}(H,\varepsilon)^{1/(H-\varepsilon)}\\|B^{H}\\|_{0,T;H-\varepsilon}^{1/(H-\varepsilon)}\\|\sigma\\|_{\infty}\\|\sigma^{\prime}\\|_{\infty}^{1/(H-\varepsilon)-1}(t-s)+2\\|b\\|_{\infty}(t-s).$	(2.12)

The desired result follows from (2.2) and (2.2), and the proof is concluded. ∎

For the case where the coefficient functions $\sigma$ or $b$ are unbounded, we have the following estimation.

Theorem 2.2.

Suppose that the coefficient functions $\sigma$ and $b$ have bounded derivatives which are Hölder continuous of order greater than $1/(H-\varepsilon)-1$ for some $\varepsilon\in(0,H-\frac{1}{2})$ . Moreover, if $\|\sigma^{\prime}\|_{\infty}+\|b^{\prime}\|_{\infty}>0$ , then for all $T>0$ , for all $0\leq s<t\leq T$ ,

\displaystyle|X_{t}-X_{s}|\leq C(H,\varepsilon,\sigma,b,T)^{1+(\|B^{H}\|_{s,t;H-\varepsilon})^{1/(H-\varepsilon)}}(|X_{0}|+1)(t-s)^{H-\varepsilon},

(2.13)

where $C(H,\varepsilon,\sigma,b,T)$ is a constant depending only on $(H,\varepsilon,\sigma,b,T)$ .

Proof.

Obviously the function $g(t)=t$ is Hölder continuous of any order $\beta\in(0,1]$ with the Hölder norm $\|g\|_{0,T;\beta}=T^{1-\beta}$ . Hence, by [Hu2007, Theorem 2 (i)], we have

	$\displaystyle\sup_{t\in[0,T]}\|X_{t}\|$	$\displaystyle\leq 2^{1+C(H,\varepsilon)T\left[\\|\sigma^{\prime}\\|_{\infty}+\\|b^{\prime}\\|_{\infty}+\|\sigma(0)\|+\|b(0)\|\right]^{1/(H-\varepsilon)}\left(\\|B^{H}\\|_{0,T;H-\varepsilon}+T^{1-(H-\varepsilon)}\right)^{1/(H-\varepsilon)}}(\|X_{0}\|+1)$
		$\displaystyle\leq C(H,\varepsilon,b,\sigma,T)^{1+(\\|B^{H}\\|_{0,t;H-\varepsilon})^{1/(H-\varepsilon)}}(\|X_{0}\|+1).$		(2.14)

In this proof, $C(H,\varepsilon)$ and $C(H,\varepsilon,b,\sigma,T)$ are generic positive constants depending only on $(H,\varepsilon)$ and $(H,\varepsilon,b,\sigma,T)$ , respectively, and they may vary in different places.

The estimations (2.5) and (2.2) are still valid. Instead of (2.2), we have

$\displaystyle\left\|D_{s+}^{\alpha}\sigma(X_{r})\right\|$	$\displaystyle=\dfrac{1}{\Gamma(1-\alpha)}\left\|\dfrac{\sigma(X_{r})}{(r-s)^{\alpha}}+\alpha\int_{s}^{r}\dfrac{\sigma(X_{r})-\sigma(X_{u})}{(r-u)^{\alpha+1}}du\right\|$
	$\displaystyle\leq\dfrac{1}{\Gamma(1-\alpha)}\dfrac{\left\|\sigma(X_{r})-\sigma(0)\right\|+\|\sigma(0)\|}{\|r-s\|^{\alpha}}+\dfrac{\alpha}{\Gamma(1-\alpha)}\int_{s}^{r}\dfrac{\left\|\sigma(X_{r})-\sigma(X_{u})\right\|}{\|r-u\|^{\alpha+1}}du$
	$\displaystyle\leq\dfrac{1}{\Gamma(1-\alpha)}\dfrac{\\|\sigma^{\prime}\\|_{\infty}\|X_{r}\|+\|\sigma(0)\|}{\|r-s\|^{\alpha}}+\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}}{\Gamma(1-\alpha)}\int_{s}^{r}\dfrac{\left\|X_{r}-X_{u}\right\|}{\|r-u\|^{\alpha+1}}du$
	$\displaystyle\leq\dfrac{1}{\Gamma(1-\alpha)}\dfrac{\\|\sigma^{\prime}\\|_{\infty}\|X_{r}\|+\|\sigma(0)\|}{\|r-s\|^{\alpha}}+\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)}\int_{s}^{r}\dfrac{1}{\|r-u\|^{\alpha+1-(H-\varepsilon)}}du$
	$\displaystyle\leq\dfrac{1}{\Gamma(1-\alpha)}\dfrac{\\|\sigma^{\prime}\\|_{\infty}\|X_{r}\|+\|\sigma(0)\|}{\|r-s\|^{\alpha}}+\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)(H-\varepsilon-\alpha)}\|r-s\|^{H-\varepsilon-\alpha}.$	(2.15)

By (2.5), (2.2), (2.2) and (2.2), we have

		$\displaystyle\left\|\int_{s}^{t}\sigma(X_{r})\circ dB_{r}^{H}\right\|$
		$\displaystyle\leq\int_{s}^{t}\left(\dfrac{1}{\Gamma(\alpha)}+\dfrac{(1-\alpha)}{(H-\varepsilon+\alpha-1)\Gamma(\alpha)}\right)\\|B^{H}\\|_{s,t;H-\varepsilon}\|t-r\|^{H-\varepsilon-1+\alpha}$
		$\displaystyle\quad\left(\dfrac{1}{\Gamma(1-\alpha)}\dfrac{\\|\sigma^{\prime}\\|_{\infty}\|X_{r}\|+\|\sigma(0)\|}{\|r-s\|^{\alpha}}+\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)(H-\varepsilon-\alpha)}\|r-s\|^{H-\varepsilon-\alpha}\right)dr$
		$\displaystyle\leq\dfrac{(H-\varepsilon)\\|B^{H}\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)\Gamma(\alpha)(H-\varepsilon+\alpha-1)}\left(\\|\sigma^{\prime}\\|_{\infty}\sup_{r\in[0,T]}\|X_{r}\|+\|\sigma(0)\|\right)\int_{s}^{t}\|t-r\|^{H-\varepsilon-1+\alpha}\|r-s\|^{-\alpha}dr$
		$\displaystyle\quad+\dfrac{\alpha(H-\varepsilon)\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)\Gamma(\alpha)(H-\varepsilon-\alpha)(H-\varepsilon+\alpha-1)}\\|B^{H}\\|_{s,t;H-\varepsilon}\int_{s}^{t}\|t-r\|^{H-\varepsilon-1+\alpha}\|r-s\|^{H-\varepsilon-\alpha}dr$
		$\displaystyle=\dfrac{(H-\varepsilon)\\|B^{H}\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)\Gamma(\alpha)(H-\varepsilon+\alpha-1)}\left(\\|\sigma^{\prime}\\|_{\infty}\sup_{r\in[0,T]}\|X_{r}\|+\|\sigma(0)\|\right)\|t-s\|^{H-\varepsilon}\beta(H-\varepsilon+\alpha,1-\alpha)$
		$\displaystyle\quad+\dfrac{\alpha(H-\varepsilon)\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)\Gamma(\alpha)(H-\varepsilon-\alpha)(H-\varepsilon+\alpha-1)}\\|B^{H}\\|_{s,t;H-\varepsilon}\|t-s\|^{2H-2\varepsilon}\beta(H-\varepsilon+\alpha,H-\varepsilon-\alpha+1)$
		$\displaystyle\leq C(H,\varepsilon)\\|B^{H}\\|_{s,t;H-\varepsilon}\|t-s\|^{H-\varepsilon}\left[\\|\sigma^{\prime}\\|_{\infty}\sup_{r\in[0,T]}\|X_{r}\|+\|\sigma(0)\|+\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}\|t-s\|^{H-\varepsilon}\right].$		(2.16)

Similarly, it is easy to see that

	$\displaystyle\left\|\int_{s}^{t}b(X_{r})dr\right\|$	$\displaystyle\leq\|t-s\|\left[\\|b^{\prime}\\|_{\infty}\sup_{r\in[0,T]}\|X_{r}\|+\|b(0)\|\right]$
		$\displaystyle\leq\|t-s\|^{H-\varepsilon}T^{1-(H-\varepsilon)}\left[\\|b^{\prime}\\|_{\infty}\sup_{r\in[0,T]}\|X_{r}\|+\|b(0)\|\right].$		(2.17)

Hence, by (2.2) and (2.2), there exists a constant $C_{0}(H,\varepsilon)$ such that

	$\displaystyle\left\|X_{t}-X_{s}\right\|$	$\displaystyle\leq\left\|\int_{s}^{t}\sigma(X_{r})\circ dB_{r}^{H}\right\|+\left\|\int_{s}^{t}b(X_{r})dr\right\|$
		$\displaystyle\leq C_{0}(H,\varepsilon)\left(\\|B^{H}\\|_{s,t;H-\varepsilon}+T^{1-(H-\varepsilon)}\right)\|t-s\|^{H-\varepsilon}\Big{[}\left(\\|\sigma^{\prime}\\|_{\infty}+\\|b^{\prime}\\|_{\infty}\right)\sup_{r\in[0,T]}\|X_{r}\|$
		$\displaystyle\quad+\|\sigma(0)\|+\|b(0)\|+\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}\|t-s\|^{H-\varepsilon}\Big{]},$

and consequently,

	$\displaystyle\\|X\\|_{s,t;H-\varepsilon}$	$\displaystyle\leq C_{0}(H,\varepsilon)\left(\\|B^{H}\\|_{s,t;H-\varepsilon}+T^{1-(H-\varepsilon)}\right)\Big{[}\left(\\|\sigma^{\prime}\\|_{\infty}+\\|b^{\prime}\\|_{\infty}\right)\sup_{r\in[0,T]}\|X_{r}\|$
		$\displaystyle\quad+\|\sigma(0)\|+\|b(0)\|+\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}\|t-s\|^{H-\varepsilon}\Big{]}.$		(2.18)

Fix a positive constant $\Delta$ such that

\displaystyle\Delta\leq\left(\dfrac{1}{3C_{0}(H,\varepsilon)\|\sigma^{\prime}\|_{\infty}(\|B^{H}\|_{0,T;H-\varepsilon}+T^{1-(H-\varepsilon)})}\right)^{1/(H-\varepsilon)}\,.

If $t-s\leq\Delta$ , we have

\displaystyle\|X\|_{s,t;H-\varepsilon}\leq\dfrac{3}{2}C_{0}(H,\varepsilon)\left(\|B^{H}\|_{0,T;H-\varepsilon}+T^{1-(H-\varepsilon)}\right)\left(|\sigma(0)|+|b(0)|+\left(\|\sigma^{\prime}\|_{\infty}+\|b^{\prime}\|_{\infty}\right)\sup_{t\in[0,T]}|X_{t}|\right).

(2.19)

Then by (2.2) and (2.19), we have

\displaystyle\|X\|_{s,t;H-\varepsilon}\leq C(H,\varepsilon,\sigma,b,T)^{1+(\|B^{H}\|_{s,t;H-\varepsilon})^{1/(H-\varepsilon)}}(|X_{0}|+1).

(2.20)

If $t-s>\Delta$ , Similar to the proof of Theorem 2.1, we divide the interval $[s,t]$ into $n=[(t-s)/\Delta]+1$ subintervals, whose lengths are smaller than $\Delta$ . Let $s=u_{0}<u_{1}<\cdots<u_{n}=t$ be endpoints of the subintervals. Then by (2.2) and (2.19), we have

$\displaystyle\|X_{t}-X_{s}\|$	$\displaystyle\leq\sum_{i=1}^{n}\|X_{u_{i}}-X_{u_{i-1}}\|$
	$\displaystyle\leq C(H,\varepsilon,\sigma,b,T)^{1+(\\|B^{H}\\|_{s,t;H-\varepsilon})^{1/(H-\varepsilon)}}(\|X_{0}\|+1)\sum_{i=1}^{n}(u_{i}-u_{i-1})^{H-\varepsilon}$
	$\displaystyle\leq C(H,\varepsilon,\sigma,b,T)^{1+(\\|B^{H}\\|_{s,t;H-\varepsilon})^{1/(H-\varepsilon)}}(\|X_{0}\|+1)n\Delta^{H-\varepsilon}$
	$\displaystyle\leq C(H,\varepsilon,\sigma,b,T)^{1+(\\|B^{H}\\|_{s,t;H-\varepsilon})^{1/(H-\varepsilon)}}(\|X_{0}\|+1)(t-s)\varDelta^{H-\varepsilon-1}$
	$\displaystyle\leq C(H,\varepsilon,\sigma,b,T)^{1+(\\|B^{H}\\|_{s,t;H-\varepsilon})^{1/(H-\varepsilon)}}(\|X_{0}\|+1)(t-s)^{H-\varepsilon}.$	(2.21)

The desired result (2.13) come from (2.20) and (2.2). ∎

Combining Theorem 2.1, Theorem 2.2, and Lemma 2.1 with dominated convergence theorem, one can prove the following corollary.

Corollary 2.1.

Assume that $\mathbb{E}[|X_{0}|^{p}]<\infty$ for $p\geq 2$ . If the conditions in Theorem 2.1 or in Theorem 2.2 hold, then $\mathbb{E}[|X_{t}|^{p}]$ is a continuous function of $t$ .

3 High-dimensional limit for Wigner-type matrices

3.1 Relative compactness of empirical spectral measure-valued processes

Denote by $\mathbf{P}(\mathbb{R})$ the space of probability measures equipped with weak topology, then $\mathbf{P}(\mathbb{R})$ is a Polish space. For $T>0$ , let $C([0,T],\mathbf{P}(\mathbb{R}))$ be the space of continuous $\mathbf{P}(\mathbb{R})$ -valued processes. In this subsection, under proper conditions, we obtain the almost sure relative compactness in $C([0,T],\mathbf{P}(\mathbb{R}))$ for the set $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ of empirical spectral measures of $Y^{N}(t)$ with entries given in (1.2) by using Lemma B.4([Anderson2010, Lemma 4.3.13]).

Theorem 3.1.

Assume one of the following hypotheses holds:

H1.

Conditions in Theorem 2.1 hold;
H2.

Conditions in Theorem 2.2 hold and $\mathbb{E}[|X_{0}|^{4}]<\infty$ .

Assume that there exists a positive function $\varphi(x)\in C^{1}(\mathbb{R})$ with bounded derivative, such that $\lim_{|x|\rightarrow\infty}\varphi(x)=\infty$ and

\displaystyle\sup_{N\in\mathbb{N}}\langle\varphi,L_{N}(0)\rangle\leq C_{0},

for some positive constant $C_{0}$ almost surely. Then for any $T>0$ , the sequence $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ is relatively compact in $C([0,T],\mathbf{P}(\mathbb{R}))$ almost surely.

Proof.

Under hypothesis H1 (hypothesis H2, resp.), by Theorem 2.1 (Theorem 2.2, resp.), we have

\displaystyle\quad|X_{t}-X_{s}|\leq\xi|t-s|^{H-\varepsilon}

where $\xi$ is a random variable with finite moment of any order (with finite $4$ -th order moment, resp.).

By (1.3) and Lemma A.1, for any $f\in C^{1}(\mathbb{R})$ with bounded derivative,

		$\displaystyle\quad\left\|\langle f,L_{N}(t)\rangle-\langle f,L_{N}(s)\rangle\right\|^{2}=\left\|\dfrac{1}{N}\sum_{i=1}^{N}\left(f(\lambda_{i}^{N}(t))-f(\lambda_{i}^{N}(s))\right)\right\|^{2}$
		$\displaystyle\leq\dfrac{1}{N}\sum_{i=1}^{N}\left\|f(\lambda_{i}^{N}(t))-f(\lambda_{i}^{N}(s))\right\|^{2}\leq\dfrac{1}{N}\\|f^{\prime}\\|_{\infty}^{2}\sum_{i=1}^{N}\left\|\lambda_{i}^{N}(t)-\lambda_{i}^{N}(s)\right\|^{2}$
		$\displaystyle\leq\dfrac{1}{N}\\|f^{\prime}\\|_{\infty}^{2}\mathrm{Tr}\left(Y^{N}(t)-Y^{N}(s)\right)^{2}=\dfrac{1}{N}\\|f^{\prime}\\|_{\infty}^{2}\sum_{i,j=1}^{N}\left(Y_{ij}^{N}(t)-Y_{ij}^{N}(s)\right)^{2}$
		$\displaystyle=\\|f^{\prime}\\|_{\infty}^{2}\left[\dfrac{2}{N^{2}}\sum_{i<j}\left(X_{ij}^{N}(t)-X_{ij}^{N}(s)\right)^{2}+\dfrac{2}{N^{2}}\sum_{i=1}^{N}\left(X_{ii}^{N}(t)-X_{ii}^{N}(s)\right)^{2}\right]$
		$\displaystyle\leq\\|f^{\prime}\\|_{\infty}^{2}(t-s)^{2H-2\varepsilon}\left(\dfrac{2}{N^{2}}\sum_{i<j}\xi_{ij}^{2}+\dfrac{2}{N^{2}}\sum_{i=1}^{N}\xi_{ii}^{2}\right)$
		$\displaystyle\leq\frac{4}{N(N+1)}\\|f^{\prime}\\|_{\infty}^{2}(t-s)^{2H-2\varepsilon}\sum_{i\leq j}\xi_{ij}^{2},$		(3.1)

where $\{\xi_{ij}\}$ are i.i.d copies of $\xi$ .

Recall that by the Arzela-Ascoli Theorem (or Lemma B.3), for any number $M>0$ , the set

\displaystyle\bigcap_{n=1}^{\infty}\left\{g\in C([0,T],\mathbb{R}):\sup_{s,t\in[0,T],|t-s|\leq\eta_{n}}|g(t)-g(s)|\leq\varepsilon_{n},\quad\sup_{t\in[0,T]}|g(t)|\leq M\right\},

where $\{\varepsilon_{n},n\in\mathbb{N}\}$ and $\{\eta_{n},n\in\mathbb{N}\}$ are two sequences of positive real numbers going to zero as $n$ goes to infinity, is compact in $C([0,T],\mathbb{R})$ .

Define

\displaystyle K=\left\{\nu\in\mathbf{P}(\mathbb{R}):\int\varphi(x)\nu(dx)\leq C_{0}+M_{0}\right\},

where $M_{0}$ is a positive number that will be determined later. Then by Lemma B.2, $K$ is compact in $\mathbf{P}(\mathbb{R})$ . Note that there exists a sequence of $C_{b}^{1}(\mathbb{R})$ functions $\{f_{i}\}_{i\in\mathbb{N}}$ that it is dense in $C_{0}(\mathbb{R})$ . Choose a positive integer $p_{0}$ , such that $p_{0}(H-\varepsilon)>1$ , and define

\displaystyle C_{T}(f_{i})=\bigcap_{n=1}^{\infty}\left\{\nu\in C([0,T],\mathbf{P}(\mathbb{R})):\sup_{s,t\in[0,T],|t-s|\leq n^{-p_{0}}}|\langle f_{i},\nu_{t}\rangle-\langle f_{i},\nu_{s}\rangle|\leq M_{i}n^{1-p_{0}(H-\varepsilon)}\right\},

where $\{M_{i}\}_{i\in\mathbb{N}}$ is a sequence of positive numbers that are independent of $n$ and will be determined later. Denote

\displaystyle\mathfrak{C}=\left\{\nu\in C([0,T],\mathbf{P}(\mathbb{R})):\nu_{t}\in K,\forall t\in[0,T]\right\}\cap\bigcap_{i=1}^{\infty}C_{T}(f_{i}),

then $\mathfrak{C}$ is compact in $C([0,T],\mathbf{P}(\mathbb{R}))$ , according to Lemma B.4 ([Anderson2010, Lemma 4.3.13]). Hence, it is enough to show

\displaystyle\mathbb{P}\left(\liminf_{N\rightarrow\infty}\big{\{}L_{N}\in\mathfrak{C}\big{\}}\right)=1.

Note that $\xi_{ij}^{2}-\mathbb{E}[\xi_{ij}^{2}]:=\xi_{ij}^{2}-m$ are i.i.d. with mean zero and finite variance denoted by $\sigma^{2}$ for $1\leq i\leq j\leq N$ . By (3.1) and the Markov inequality, we have

		$\displaystyle\quad\mathbb{P}\left(L_{N}\notin\Big{\{}\nu\in C([0,T],\mathbf{P}(\mathbb{R})):\nu_{t}\in K,\forall t\in[0,T]\Big{\}}\right)$
		$\displaystyle=\mathbb{P}\Big{(}\exists t\in[0,T],L_{N}(t)\notin K\Big{)}=\mathbb{P}\left(\sup_{t\in[0,T]}\langle\varphi,L_{N}(t)\rangle>C_{0}+M_{0}\right)$
		$\displaystyle\leq\mathbb{P}\left(\sup_{t\in[0,T]}\left\|\langle\varphi,L_{N}(t)\rangle-\langle\varphi,L_{N}(0)\rangle\right\|>M_{0}\right)$
		$\displaystyle\leq\mathbb{P}\left(\frac{4}{N(N+1)}\\|\varphi^{\prime}\\|_{\infty}^{2}T^{2H-2\varepsilon}\sum_{i\leq j}\xi_{ij}^{2}>M_{0}^{2}\right)$
		$\displaystyle=\mathbb{P}\left(\dfrac{2}{N(N+1)}\sum_{i\leq j}\xi_{ij}^{2}-m>\dfrac{M_{0}^{2}}{2\\|\varphi^{\prime}\\|_{\infty}^{2}T^{2H-2\varepsilon}}-m\right)$
		$\displaystyle\leq\left(\dfrac{M_{0}^{2}}{2\\|\varphi^{\prime}\\|_{\infty}^{2}T^{2H-2\varepsilon}}-m\right)^{-2}\mathbb{E}\left[\left(\dfrac{2}{N(N+1)}\sum_{i\leq j}\xi_{ij}^{2}-m\right)^{2}\right]$
		$\displaystyle=\left(\dfrac{M_{0}^{2}}{2\\|\varphi^{\prime}\\|_{\infty}^{2}T^{2H-2\varepsilon}}-m\right)^{-2}\dfrac{4}{N^{2}(N+1)^{2}}\mathbb{E}\left[\left(\sum_{i\leq j}\big{[}\xi_{ij}^{2}-m\big{]}\right)^{2}\right]$
		$\displaystyle=\left(\dfrac{M_{0}^{2}}{2\\|\varphi^{\prime}\\|_{\infty}^{2}T^{2H-2\varepsilon}}-m\right)^{-2}\dfrac{4}{N^{2}(N+1)^{2}}\mathbb{E}\left[\sum_{i\leq j}\big{(}\xi_{ij}^{2}-m\big{)}^{2}\right]$
		$\displaystyle=\left(\dfrac{M_{0}^{2}}{2\\|\varphi^{\prime}\\|_{\infty}^{2}T^{2H-2\varepsilon}}-m\right)^{-2}\dfrac{2}{N(N+1)}\sigma^{2}.$		(3.2)

If we choose $M_{0}=2\|\varphi^{\prime}\|_{\infty}T^{H-\varepsilon}(1+m)$ , then (3.1) becomes

		$\displaystyle\quad\mathbb{P}\left(L_{N}\notin\Big{\{}\nu\in C([0,T],\mathbf{P}(\mathbb{R})):\nu_{t}\in K,\forall t\in[0,T]\Big{\}}\right)$
		$\displaystyle\leq\dfrac{2\sigma^{2}}{N(N+1)(2m^{2}+3m+2)^{2}}\leq\dfrac{8\sigma^{2}}{N(N+1)}.$		(3.3)

Similarly, by (3.1) and the Markov inequality, we have

		$\displaystyle\quad\mathbb{P}\left(L_{N}\notin C_{T}(f_{i})\right)$
		$\displaystyle\leq\sum_{n=1}^{\infty}\mathbb{P}\left(L_{N}\notin\left\{\nu\in C([0,T],\mathbf{P}(\mathbb{R})):\sup_{s,t\in[0,T],\|t-s\|\leq n^{-p_{0}}}\|\langle f_{i},\nu_{t}\rangle-\langle f_{i},\nu_{s}\rangle\|\leq M_{i}n^{1-p_{0}(H-\varepsilon)}\right\}\right)$
		$\displaystyle=\sum_{n=1}^{\infty}\mathbb{P}\left(\sup_{s,t\in[0,T],\|t-s\|\leq n^{-p_{0}}}\|\langle f_{i},L_{N}(t)\rangle-\langle f_{i},L_{N}(s)\rangle\|>M_{i}n^{1-p_{0}(H-\varepsilon)}\right)$
		$\displaystyle\leq\sum_{n=1}^{\infty}\mathbb{P}\left(2\\|f_{i}^{\prime}\\|_{\infty}^{2}n^{-p_{0}(2H-2\varepsilon)}\left(\dfrac{2}{N(N+1)}\sum_{i\leq j}\xi_{ij}^{2}\right)>M_{i}^{2}n^{2-2p_{0}(H-\varepsilon)}\right)$
		$\displaystyle=\sum_{n=1}^{\infty}\mathbb{P}\left(\dfrac{2}{N(N+1)}\sum_{i\leq j}\xi_{ij}^{2}-m>\dfrac{M_{i}^{2}n^{2}}{2\\|f_{i}^{\prime}\\|_{\infty}^{2}}-m\right)$
		$\displaystyle\leq\sum_{n=1}^{\infty}\left(\dfrac{M_{i}^{2}n^{2}}{2\\|f_{i}^{\prime}\\|_{\infty}^{2}}-m\right)^{-2}\mathbb{E}\left[\left(\dfrac{2}{N(N+1)}\sum_{i\leq j}\xi_{ij}^{2}-m\right)^{2}\right]$
		$\displaystyle=\sum_{n=1}^{\infty}\left(\dfrac{M_{i}^{2}n^{2}}{2\\|f_{i}^{\prime}\\|_{\infty}^{2}}-m\right)^{-2}\dfrac{4}{N^{2}(N+1)^{2}}\mathbb{E}\left[\sum_{i\leq j}(\xi_{ij}^{2}-m)^{2}\right]$
		$\displaystyle\leq\sum_{n=1}^{\infty}\left(\dfrac{M_{i}^{2}n^{2}}{2\\|f_{i}^{\prime}\\|_{\infty}^{2}}-m\right)^{-2}\dfrac{2}{N(N+1)}\sigma^{2}.$		(3.4)

Now, we choose $M_{i}=2i\|f_{i}^{\prime}\|_{\infty}\gamma$ , where $\gamma$ is a positive real number such that $2\gamma^{2}>m$ . Then (3.1) becomes

\displaystyle\mathbb{P}\left(L_{N}\notin C_{T}(f_{i})\right)

\displaystyle\leq\sum_{n=1}^{\infty}\dfrac{2\sigma^{2}}{N(N+1)(2i^{2}n^{2}{\gamma}^{2}-m)^{2}}.

(3.5)

Hence, by the definition of $\mathfrak{C}$ , (3.1) and (3.5), we have

$\displaystyle\sum_{N=1}^{\infty}\mathbb{P}\left(L_{N}\notin\mathfrak{C}\right)$	$\displaystyle\leq\sum_{N=1}^{\infty}\mathbb{P}\left(L_{N}\notin\left\{\nu\in C([0,T],\mathbf{P}(\mathbb{R})):\nu_{t}\in K,\forall t\in[0,T]\right\}\right)$
	$\displaystyle\qquad+\sum_{N=1}^{\infty}\sum_{i=1}^{\infty}\mathbb{P}\left(L_{N}(t)\notin C_{T}(f_{i})\right)$
	$\displaystyle\leq\sum_{N=1}^{\infty}\dfrac{8\sigma_{D}^{2}}{N(N+1)}+\sum_{N=1}^{\infty}\sum_{i=1}^{\infty}\sum_{n=1}^{\infty}\dfrac{2\sigma^{2}}{N(N+1)(2i^{2}n^{2}\gamma^{2}-m)^{2}}$
	$\displaystyle<\infty.$	(3.6)

Therefore, by the Borel-Cantelli Lemma and (3.1), we have

\displaystyle\mathbb{P}\left(\limsup_{N\rightarrow\infty}\left\{L_{N}\notin\mathfrak{C}\right\}\right)=0.

The proof is concluded. ∎

3.2 Limit of empirical spectral distributions

Recall that the celebrated semicircle distribution $\mu_{sc}(dx)$ on $[-2,2]$ has density function

\displaystyle p_{sc}(x)=\dfrac{\sqrt{4-x^{2}}}{2\pi}1_{[-2,2]}(x).

(3.7)

Theorem 3.2.

Suppose that the conditions in Theorem 3.1 hold. We also assume that $\mathbb{E}[|X_{0}|^{2}]<\infty$ and denote $m_{t}=\mathbb{E}[X_{t}]$ and $d_{t}=(\mathbb{E}[|X_{t}|^{2}]-m_{t}^{2})^{1/2}$ . Then for any $T>0$ , the sequence $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ converges to $\mu=\{\mu_{t},t\in[0,T]\}$ in $C([0,T],\mathbf{P}(\mathbb{R}))$ almost surely, as $N\to\infty$ . The limit measure $\mu_{t}$ has density $p_{t}(x)=p_{sc}(x/d_{t})/d_{t}$ , where $p_{sc}(x)$ is given by (3.7).

Proof.

First, for fixed $t\in[0,T]$ , we prove the almost sure weak convergence of the empirical measure $L_{N}(t)$ .

Note that by Corollary 2.1, $m_{t}$ and $d_{t}$ are continuous functions of $t$ on $[0,T]$ . Let $\widetilde{Y}^{N}(t)=\left(\widetilde{Y}_{ij}^{N}(t)\right)_{1\leq i,j\leq N}$ be a symmetric matrix with entries

\displaystyle\widetilde{Y}_{ij}^{N}(t)=\dfrac{Y_{ij}^{N}(t)-m_{t}/\sqrt{N}}{d_{t}},~{}~{}1\leq i\leq j\leq N.

(3.8)

Then $\widetilde{Y}^{N}(t)$ is a Wigner matrix. Let $\tilde{\lambda}_{1}^{N}(t)\leq\cdots\leq\tilde{\lambda}_{N}^{N}(t)$ be the eigenvalues of $\widetilde{Y}^{N}(t)$ and $\widetilde{L}_{N}(t)(dx)=\sum_{i=1}^{N}\delta_{\tilde{\lambda}_{i}^{N}(t)}(dx)/N$ be the empirical spectral measure. By Lemma A.2, $\widetilde{L}_{N}(t)(dx)$ converges weakly to the semicircle distribution $p_{sc}(x)dx$ almost surely for all $0\leq t\leq T$ . Hence the empirical measure of the eigenvalues of $d_{t}\widetilde{Y}^{N}(t)$ converges weakly to $\frac{1}{d_{t}}p_{sc}\left(\frac{x}{d_{t}}\right)dx$ almost surely.

Note that by (3.8), $Y^{N}(t)=d_{t}\widetilde{Y}^{N}(t)+\dfrac{m_{t}}{\sqrt{N}}E_{N}$ , where $E_{N}$ is an $N\times N$ matrix with unit entries. Then by [Tao2012, Exercise 2.4.4], we can conclude that the empirical distribution $L_{N}(t)(dx)$ converges weakly to $\dfrac{1}{d_{t}}p_{sc}\left(\dfrac{x}{d_{t}}\right)dx$ , almost surely, for all $t\in[0,T]$ .

Now, we show that $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ converges weakly to $\{\mu_{t},t\in[0,T]\}_{N\in\mathbb{N}}$ almost surely.

Let $\{L_{N_{i}}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ be an arbitrary convergent subsequence with the limit $\{\mu_{t},t\in[0,T]\}$ , then for $t\in[0,T]$ , $\mu_{t}(dx)=p_{sc}(x/d_{t})/d_{t}dx$ . Therefore, noting that Theorem 3.1 yields that the sequence $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ is relatively compact almost surely in $C([0,T],\mathbf{P}(\mathbb{R}))$ , any subsequence of $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ has a convergent subsequence with the unique limit $\{\mu_{t}(dx)=p_{sc}(x/d_{t})/d_{t}dx,t\in[0,T]\}$ . This implies that the total sequence $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ converges in $C([0,T],\mathbf{P}(\mathbb{R}))$ with the limit $\{\mu_{t}(dx)=p_{sc}(x/d_{t})/d_{t}dx,t\in[0,T]\}$ , almost surely. ∎

Remark 3.1.

If $\sigma(x)=1$ , $b(x)=0$ and $X_{0}=0$ , then the solution to the SDE (1.1) is the fractional Brownian motion $X_{t}=B_{t}^{H}$ , and Theorem 3.2 implies that the the empirical spectral measure converges weakly to the scaled semicircle distribution with density $p_{t}(x)=p_{sc}(x/t^{H})/t^{H}$ almost surely. This improves the results in [Pardo2016], where the convergence of the empirical spectral measure in law is obtained.

Remark 3.2.

The Stieltjes transform $G_{t}(z)$ of the limiting measure $\mu_{t}$ is

\displaystyle G_{t}(z)=\int\dfrac{\mu_{t}(dx)}{z-x}=\int\dfrac{p_{sc}(x/{d_{t}})/d_{t}}{z-x}dx=\int\dfrac{p_{sc}(x)}{z-d_{t}x}dx=\dfrac{1}{d_{t}}G_{sc}(z/d_{t}),

where

\displaystyle G_{sc}(z)=\int\dfrac{p_{sc}(x)}{z-x}dx,

is the Stieltjes transform of the semi-circle distribution. If we assume that the variance $d_{t}^{2}$ of the solution $X_{t}$ is continuously differentiable on $(0,T)$ , then we have

$\displaystyle\partial_{t}G_{t}(z)$	$\displaystyle=\partial_{t}\int\dfrac{p_{sc}(x)}{z-d_{t}x}dx=d_{t}^{\prime}\int\dfrac{xp_{sc}(x)}{(z-d_{t}x)^{2}}dx$
	$\displaystyle=\dfrac{d_{t}^{\prime}}{d_{t}}\int\left(-\dfrac{1}{z-d_{t}x}+\dfrac{z}{(z-d_{t}x)^{2}}\right)p_{sc}(x)dx$
	$\displaystyle=-\dfrac{d_{t}^{\prime}}{d_{t}}G_{t}(z)-\dfrac{d_{t}^{\prime}}{d_{t}}z\partial_{z}G_{t}(z).$	(3.9)

By [Bai2010, Lemma 2.11], it is easy to get

\displaystyle d_{t}^{2}G_{t}(z)^{2}=zG_{t}(z)-1,

and by (3.9),

	$\displaystyle\partial_{t}G_{t}(z)$	$\displaystyle=-\dfrac{d_{t}^{\prime}}{d_{t}}(G_{t}(z)+z\partial_{z}G_{t}(z))=-\dfrac{d_{t}^{\prime}}{d_{t}}\partial_{z}(zG_{t}(z)-1)$
		$\displaystyle=-d_{t}d_{t}^{\prime}\partial_{z}(G_{t}(z)^{2})=-(d_{t}^{2})^{\prime}G_{t}(z)\partial_{z}G_{t}(z).$		(3.10)

For the case $X_{t}=B_{t}^{H}$ with $d_{t}^{2}=t^{2H}$ , equation (3.2) becomes

\displaystyle\partial_{t}G_{t}(z)=-2Ht^{2H-1}G_{t}(z)\partial_{z}G_{t}(z).

(3.11)

Denoting $F_{t}(z)=G_{t^{1/2H}}(z)$ , by change of variable and (3.11), one can deduce that $F_{t}(z)$ satisfies the complex Burgers’ equation

\displaystyle\partial_{t}F_{t}(z)=-F_{t}(z)\partial_{z}F_{t}(z).

This relationship was obtained in [Jaramillo2019].

3.3 Complex case

In this subsection, we consider the following 2-dimensional SDE for $Z_{t}=(Z_{t}^{(1)},Z_{t}^{(2)})$ ,

\displaystyle dZ_{t}=\tilde{\sigma}(Z_{t})\circ dB_{t}^{H}+\tilde{b}(Z_{t})dt,

(3.12)

with initial value $Z_{0}$ that is independent of $B_{t}^{H}$ . Here $\tilde{b}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ , $\tilde{\sigma}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2\times 2}$ are continuously differentiable functions and $B_{t}^{H}$ is a 2-dimensional fractional Brownian motion. By [Lyons1994], there exists a unique solution to SDE (3.12), if $\tilde{\sigma}$ and $\tilde{b}$ have bounded derivatives which are Hölder continuous of order greater than $1/H-1$ .

Denote by $\iota$ the imaginary unit. Let $\{Z_{kl}\}_{k,l\geq 1}$ be i.i.d. copies of $Z_{t}^{(1)}+\iota Z_{t}^{(2)}$ and $Z^{N}(t)=\left(Z_{kl}^{N}(t)\right)_{1\leq k,l\leq N}$ be a Hermitian $N\times N$ matrix with entries

\displaystyle Z_{kl}^{N}(t)=\begin{cases}\dfrac{1}{\sqrt{N}}Z_{kl}(t),&1\leq k<l\leq N,\\ \dfrac{1}{\sqrt{N}}X_{kk}(t),&1\leq k\leq N.\end{cases}

(3.13)

Here, $\{X_{kk}(t)\}$ are i.i.d. copies of the real-valued process $X_{t}$ satisfying (1.1) and independent of the family $\{Z_{kl}(t)\}_{k,l\geq 1}$ . Let $\lambda_{1}^{N}(t)\leq\cdots\leq\lambda_{N}^{N}(t)$ be the eigenvalues of $Z^{N}(t)$ and denote the empirical spectral measure by

\displaystyle L_{N}(t)(dx)=\dfrac{1}{N}\sum_{k=1}^{N}\delta_{\lambda_{k}^{N}(t)}(dx).

Theorem 3.3.

Suppose that the coefficient functions $\tilde{\sigma}$ , $\tilde{b}$ , $\sigma$ and $b$ have bounded (partial) derivatives which are Hölder continuous of order greater than $1/(H-\varepsilon)-1$ . Besides, assume that among the following conditions,

$(a_{1})$

$\tilde{\sigma}$ and $\tilde{b}$ are bounded and $\mathbb{E}[\|Z_{0}\|^{2}]<\infty$ ;
$(a_{2})$

$\|(\tilde{\sigma}_{x},\tilde{\sigma}_{y})\|_{L^{\infty}(\mathbb{R}^{2})}+\|(\tilde{b}_{x},\tilde{b}_{y})\|_{L^{\infty}(\mathbb{R}^{2})}>0$ , $\mathbb{E}[\|Z_{0}\|^{4}]<\infty$ ;
$(b_{1})$

$\sigma$ and $b$ are bounded and $\mathbb{E}[|X_{0}|^{2}]<\infty$ ;
$(b_{2})$

$\|\sigma^{\prime}\|_{\infty}+\|b^{\prime}\|_{\infty}>0$ , $\mathbb{E}[|X_{0}|^{4}]<\infty$ ;

$(a_{1})$ or $(a_{2})$ holds and $(b_{1})$ or $(b_{2})$ holds. Furthermore, suppose that there exists a positive function $\varphi(x)\in C^{1}(\mathbb{R})$ with bounded derivative, such that $\lim_{|x|\rightarrow\infty}\varphi(x)=+\infty$ and

\displaystyle\sup_{N\in\mathbb{N}}\langle\varphi,L_{N}(0)\rangle\leq C_{0},

for some positive constant $C_{0}$ almost surely.

Then for any $T>0$ , $\mathbb{E}[|X_{t}|^{2}+\|Z_{t}\|^{2}]<\infty$ for $t\in[0,T]$ , and the sequence $\{L_{N}(t),t\in[0,t]\}_{N\in\mathbb{N}}$ converges to $\mu=\{\mu_{t},t\in[0,T]\}$ in $C([0,T],\mathbf{P}(\mathbb{R}))$ almost surely. The limiting measure $\mu_{t}$ has density $p_{t}(x)=p_{sc}(x/d_{Z}(t))/d_{Z}(t)$ , where $d_{Z}^{2}(t)$ is the variance of the solution $Z_{t}$ to SDE (3.12).

Proof.

The proof is similar to the real case, which is sketched below.

First of all, following the proof of Theorem 2.1 or Theorem 2.2, we can still establish the pathwise Hölder continuity for the solution $Z_{t}=(Z_{t}^{(1)},Z_{t}^{(2)})$ to the SDE (3.12). More specifically, for the case that condition $(a_{1})$ holds, we can obtain

	$\displaystyle\quad\left\|Z_{t}^{(1)}-Z_{s}^{(1)}\right\|+\left\|Z_{t}^{(2)}-Z_{s}^{(2)}\right\|$
	$\displaystyle\leq C(H,\varepsilon)\\|\tilde{\sigma}\\|_{\infty}\left[\\|B^{H}\\|_{0,T;H-\varepsilon}(t-s)^{H-\varepsilon}\vee\\|B^{H}\\|_{0,T;H-\varepsilon}^{1/(H-\varepsilon)}(t-s)\\|\tilde{\sigma}^{\prime}\\|_{\infty}^{1/(H-\varepsilon)-1}\right]$
	$\displaystyle\quad+2\\|\tilde{b}\\|_{\infty}(t-s),$

and for the case of $(a_{2})$ ,

	$\displaystyle\quad\left\|Z_{t}^{(1)}-Z_{s}^{(1)}\right\|+\left\|Z_{t}^{(2)}-Z_{s}^{(2)}\right\|$
	$\displaystyle\leq C(H,\varepsilon,\tilde{\sigma},\tilde{b},T)^{1+(\\|B^{H}\\|_{s,t;H-\varepsilon})^{1/(H-\varepsilon)}}(\|\|Z_{0}\|\|+1)(t-s)^{H-\varepsilon},$

for all $T>0$ , for all $0\leq s<t\leq T$ . Thus, by Lemma 2.1 and the moment assumption on $Z_{0}$ , we have that for both cases, there exists a positive random variable $\zeta$ with finite second moment, such that

\displaystyle\left|Z_{t}^{(1)}-Z_{s}^{(1)}\right|^{2}+\left|Z_{t}^{(2)}-Z_{s}^{(2)}\right|^{2}\leq\zeta(t-s)^{2H-2\varepsilon},~{}~{}\text{a.s.}

(3.14)

Similar to (3.1), we have

\displaystyle\quad\left|\langle f,L_{N}(t)\rangle-\langle f,L_{N}(s)\rangle\right|^{2}\leq\dfrac{1}{N^{2}}\|f^{\prime}\|_{\infty}^{2}(t-s)^{2H-2\varepsilon}\left(\sum_{k=1}^{N}\xi_{kk}^{2}+2\sum_{k<l}\zeta_{kl}^{2}\right),

for any $f\in C^{1}(\mathbb{R})$ with bounded derivative. Then following the same approach used in the proof of Theorem 3.1, we can show that the sequence of empirical spectral measures $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ is relatively compact in $C([0,T],\mathbf{P}(\mathbb{R}))$ almost surely.

Following the proof of Corollary 2.1, it is easy to see that the mean $m_{Z}(t)=\mathbb{E}[Z_{t}]$ and the variance $d_{Z}^{2}(t)=\mathbb{E}[\|Z_{t}-m_{Z}(t)\|^{2}]$ are continuous functions of $t$ .

Next, we introduce a Hermitian matrix $\widetilde{Z}^{N}(t)=\left(\widetilde{Z}_{kl}^{N}(t)\right)_{1\leq k,l\leq N}$ satisfying

\displaystyle Z^{N}(t)=\dfrac{m_{Z}(t)}{\sqrt{N}}\left(E_{N}-I_{N}\right)+\frac{m_{t}}{\sqrt{N}}I_{N}+d_{Z}(t)\widetilde{Z}^{N}(t).

Hence, $\widetilde{Z}^{N}(t)$ is a Hermitian Wigner matrix for all $t\in[0,T]$ . Finally, by [Tao2012, Exercise 2.4.3, Exercise 2.4.4], we can conclude that the almost-sure limit of the empirical distribution of the eigenvalues of $Z^{N}(t)$ coincides with that of $d_{Z}\widetilde{Z}^{N}(t)$ for all $t\in[0,T]$ . Therefore, by Lemma A.3, $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ converges towards $\{\mu_{t},t\in[0,T]\}$ in $C([0,T],\mathbf{P}(\mathbb{R}))$ with density $\mu_{t}(dx)=p_{t}(x)dx=p_{sc}(x/d_{Z}(t))/d_{Z}(t)dx$ . ∎

Remark 3.3.

Similar to Remark 3.2, the Stieltjes transform of the limiting measure $\mu_{t}$ satisfies the differential equation (3.2) with $d_{t}$ replaced by $d_{Z}(t)$ .

4 High-dimensional limit for symmetric matrices with dependent entries

Let $\{X_{(i,j)}(t)\}_{i,j\in\mathbb{Z}}$ be i.i.d. copies of $X_{t}$ , the solution of (1.1). Fix a finite index set $I\subset\mathbb{Z}^{2}$ and a family of constants $\{a_{r}:r\in I\}$ . Let $|I|=\max\{|i_{1}-i_{2}|\vee|j_{1}-j_{2}|:(i_{1},j_{1}),(i_{2},j_{2})\in I\}$ and $\#I$ be the cardinality of $I$ . Note that $\#I\leq(2|I|+1)^{2}$ . Let $R^{N}(t)=\left(R_{ij}^{N}(t)\right)_{1\leq i,j\leq N}$ be an $N\times N$ real symmetric matrix with entries

\displaystyle R_{ij}^{N}(t)=\dfrac{1}{\sqrt{N}}\sum_{r\in I}a_{r}X_{(i,j)+r}(t),\quad 1\leq i\leq j\leq N.

(4.1)

Let $\lambda_{1}^{N}(t)\leq\cdots\leq\lambda_{N}^{N}(t)$ be the eigenvalues of $R^{N}(t)$ , and

\displaystyle L_{N}(t)(dx)=\dfrac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}^{N}(t)}(dx)

be the empirical spectral measure of $R^{N}(t)$ .

Theorem 4.1.

Suppose that the conditions in Theorem 3.1 hold. Then for any $T>0$ , the sequence $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ converges to $\{\mu_{t},t\in[0,T]\}$ in $C([0,T],\mathbf{P}(\mathbb{R}))$ almost surely. The Stieltjes transform $S_{t}(z)=\int(z-x)^{-1}\mu_{t}(dx)$ of the limit measure is given by, for $z\in\mathbb{C}\backslash\mathbb{R}$ ,

\displaystyle S_{t}(z)=\int_{0}^{1}h_{t}(x,z)dx,

where $h_{t}(x,z)$ is the solution to the equation

\displaystyle h_{t}(x,z)=\left(-z+\int_{0}^{1}f_{t}(x,y)h_{t}(y,z)dy\right)^{-1},

with

\displaystyle\quad f_{t}(x,y)=\sum_{k,l\in\mathbb{Z}}\gamma_{t}(k,l)e^{-2\pi i(kx+ly)},

where $\gamma_{t}(k,l)=\gamma_{t}(l,k)=d_{t}^{2}\sum_{r\in I\cap(I+(k,l))}a_{r}a_{r-(k,l)}$ for $k\leq l$ .

Proof.

By Theorem 2.1 and Theorem 2.2, we have the estimation

\displaystyle|X_{(i,j)}(t)-X_{(i,j)}(s)|\leq\xi_{(i,j)}|t-s|^{H-\varepsilon},\forall i,j\in\mathbb{Z},

where $\{\xi_{(i,j)}\}_{i,j\in\mathbb{Z}}$ are i.i.d. copies of $\xi$ with $\mathbb{E}|\xi|^{4}<\infty$ . Thus, for $1\leq i\leq j\leq N$ ,

	$\displaystyle\left\|R_{ij}^{N}(t)-R_{ij}^{N}(s)\right\|$	$\displaystyle\leq\dfrac{1}{\sqrt{N}}\sum_{r\in I}\|a_{r}\|\left\|X_{(i,j)+r}(t)-X_{(i,j)+r}(s)\right\|$
		$\displaystyle\leq\dfrac{1}{\sqrt{N}}\sum_{r\in I}\|a_{r}\|\xi_{(i,j)+r}(t-s)^{H-\varepsilon}.$

Define

\displaystyle F_{ij}=\left(\sum_{r\in I}|a_{r}|\xi_{(i,j)+r}\right)^{2},1\leq i\leq j.

Then all $F_{ij}$ ’s for $1\leq i\leq j$ are distributed identically with finite second moment.

Analogous to (3.1), we have

	$\displaystyle\left\|\langle f,L_{N}(t)\rangle-\langle f,L_{N}(s)\rangle\right\|^{2}$	$\displaystyle=\dfrac{1}{N}\\|f^{\prime}\\|_{\infty}^{2}\sum_{i,j=1}^{N}\left(R_{ij}^{N}(t)-R_{ij}^{N}(s)\right)^{2}$
		$\displaystyle\leq\dfrac{2}{N}\\|f^{\prime}\\|_{\infty}^{2}\sum_{i\leq j}\left(R_{ij}^{N}(t)-R_{ij}^{N}(s)\right)^{2}$
		$\displaystyle\leq\dfrac{2}{N^{2}}\\|f^{\prime}\\|_{\infty}^{2}\sum_{i\leq j}F_{ij}(t-s)^{2H-2\varepsilon}.$

Noting that the mutual independence among $\{\xi_{ij}\}$ implies the independence between $F_{ij}$ and $F_{kl}$ if $k\notin[i-|I|,i+|I|]$ or $l\notin[j-|I|,j+|I|]$ , we have

	$\displaystyle\quad\mathbb{E}\left[\left(\sum_{i\leq j}\left(F_{ij}-\mathbb{E}\left[F_{ij}\right]\right)\right)^{2}\right]=\sum_{i\leq j}\sum_{k\leq l}\mathbb{E}\left[\left(F_{ij}-\mathbb{E}\left[F_{ij}\right]\right)\left(F_{kl}-\mathbb{E}\left[F_{kl}\right]\right)\right]$
	$\displaystyle=\sum_{i\leq j}\sum_{k=i-\|I\|}^{i+\|I\|}\sum_{l=j-\|I\|}^{j+\|I\|}\mathbb{E}\left[\left(F_{ij}-\mathbb{E}\left[F_{ij}\right]\right)\left(F_{kl}-\mathbb{E}\left[F_{kl}\right]\right)\right]$
	$\displaystyle\leq\sum_{i\leq j}\sum_{k=i-\|I\|}^{i+\|I\|}\sum_{l=j-\|I\|}^{j+\|I\|}\left(\mathbb{E}\left[\left(F_{ij}-\mathbb{E}\left[F_{ij}\right]\right)^{2}\right]\mathbb{E}\left[\left(F_{kl}-\mathbb{E}\left[F_{kl}\right]\right)^{2}\right]\right)^{1/2}$
	$\displaystyle=\dfrac{(2\|I\|+1)^{2}N(N+1)}{2}\left(\mathbb{E}[F_{00}^{2}]-(E[F_{00}])^{2}\right).$

Thus, following the proof of Theorem 3.1, we may get estimations analogous to (3.1) and (3.1) therein, and then obtain the almost sure relatively compactness of the empirical spectral measure $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ .

Now, let $\widetilde{R}^{N}(t)=\left(\widetilde{R}_{ij}^{N}(t)\right)_{1\leq i,j\leq N}$ be a symmetric matrix with entries

\displaystyle\widetilde{R}_{ij}^{N}(t):=R_{ij}^{N}(t)-\mathbb{E}[R_{ij}^{N}(t)]=\dfrac{1}{\sqrt{N}}\sum_{r\in I}a_{r}\left(X_{(i,j)+r}(t)-\mathbb{E}\left[X_{(i,j)+r}(t)\right]\right),\quad 1\leq i\leq j\leq N.

Let $\widetilde{L}_{N}(t)$ be the empirical spectral measure of $\widetilde{R}^{N}(t)$ . Then by Lemma A.6 ([Banna2015, Theorem 3]), for each $t\in[0,T]$ , $\widetilde{L}_{N}(t)$ converges to a deterministic probability measure $\mu_{t}$ almost surely. Moreover, the Stieltjes transform of the limit measure $\mu_{t}$ is given by

\displaystyle S_{t}(z)=\int_{0}^{1}h_{t}(x,z)dx,

where $h_{t}(x,z)$ is the solution to the equation

\displaystyle h(x,z)=\left(-z+\int_{0}^{1}f(x,y)h(y,z)dy\right)^{-1},

with

\displaystyle\quad f(x,y)=\sum_{k,l\in\mathbb{Z}}\gamma_{k,l}e^{-2\pi i(kx+ly)},

where, for $k\leq l$ ,

	$\displaystyle\gamma_{k,l}=\gamma_{l,k}$	$\displaystyle=\mathbb{E}\left[\sum_{r\in I}a_{r}\left(X_{r}(t)-\mathbb{E}\left[X_{r}(t)\right]\right)\sum_{r^{\prime}\in I}a_{r^{\prime}}\left(X_{(k,l)+r^{\prime}}(t)-\mathbb{E}\left[X_{(k,l)+r^{\prime}}(t)\right]\right)\right]$
		$\displaystyle=\mathbb{E}\left[\sum_{r\in I}a_{r}\left(X_{r}(t)-\mathbb{E}\left[X_{r}(t)\right]\right)\sum_{r^{\prime}\in I+(k,l)}a_{r^{\prime}-(k,l)}\left(X_{r^{\prime}}(t)-\mathbb{E}\left[X_{r^{\prime}}(t)\right]\right)\right]$
		$\displaystyle=\sum_{r\in I\cap(I+(k,l))}a_{r}a_{r-(k,l)}\mathbb{E}\left[\left(X_{r}(t)-\mathbb{E}\left[X_{r}(t)\right]\right)^{2}\right]$
		$\displaystyle=d_{t}^{2}\sum_{r\in I\cap(I+(k,l))}a_{r}a_{r-(k,l)}.$

Finally, by [Tao2012, Exercise 2.4.4], the empirical spectral measure $L_{N}(t)(dx)$ of $R^{N}(t)$ converges to the same limit $\mu_{t}$ almost surely. The proof is concluded. ∎

5 High-dimensional limit for Wishart-type matrices

5.1 Real case

Recall that $\{X_{ij}(t)\}_{i,j\geq 1}$ are i.i.d. copies of $X_{t}$ which is the solution to (1.1). Let

\widehat{U}^{N}(t)=\left(\widehat{U}_{ij}^{N}(t)\right)_{1\leq i\leq p,\,1\leq j\leq N}

be a $p\times N$ matrix with entries $\widehat{U}_{ij}^{N}(t)=X_{ij}(t)-\mathbb{E}\left[X_{ij}(t)\right]$ . Here, $p=p(N)$ is a positive integer that depends on $N$ . Let

\displaystyle U^{N}(t)=\dfrac{1}{N}\widehat{U}^{N}(t)\widehat{U}^{N}(t)^{\intercal}

(5.1)

be a $p\times p$ symmetric matrix with $p$ eigenvalues $\lambda_{1}^{N}(t)\leq\cdots\leq\lambda_{p}^{N}(t)$ , and

\displaystyle L_{N}(t)(dx)=\dfrac{1}{p}\sum_{i=1}^{p}\delta_{\lambda_{i}^{N}(t)}(dx)

be the empirical spectral measure of $U^{N}(t)$ .

Theorem 5.1.

Suppose that one of the following conditions holds,

(i)

Conditions in Theorem 2.1 hold and $\mathbb{E}[|X_{0}|^{2}]<\infty$ ;
(ii)

Conditions in Theorem 2.2 hold and $\mathbb{E}[|X_{0}|^{4}]<\infty$ .

Assume that there exists a positive function $\varphi(x)\in C^{1}(\mathbb{R})$ with bounded derivative, such that $\lim_{|x|\rightarrow\infty}\varphi(x)=+\infty$ and

\displaystyle\sup_{N\in\mathbb{N}}\langle\varphi,L_{N}(0)\rangle\leq C_{0},

for some positive constant $C_{0}$ almost surely. Furthermore, assume that there exists a positive constant $c$ , such that $p/N\rightarrow c$ as $N\rightarrow\infty$ .

Then for any $T>0$ , $\mathbb{E}[|X_{t}|^{2}]<\infty$ for $t\in[0,T]$ , and the sequence $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ converges in probability to $\{\mu_{t},t\in[0,T]\}$ in $C([0,T],\mathbf{P}(\mathbb{R}))$ , where $\mu_{t}(dx)=\mu_{MP}(c,d_{t})(dx)$ with $\mu_{MP}$ given in (A.1).

Proof.

Noting that $\mathbb{E}[|X_{ij}(t)|^{2}]$ exists finitely for all $1\leq i\leq p,1\leq j\leq N$ , we have that $\widehat{U}_{ij}^{N}(t)$ has mean 0 and finite second moment $d_{t}^{2}:=\mathbb{E}[|\widehat{U}_{ij}^{N}(t)|^{2}]$ . Then by Lemma A.4, for any $t\in[0,T]$ , almost surely, the empirical distribution

\displaystyle L_{N}(t)(dx)\rightarrow\mu_{MP}(c,d_{t})(dx)

(5.2)

weakly as $N\to\infty$ . Thus, it remains to obtain the tightness of $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ in the space $C([0,T],\mathbf{P}(\mathbb{R}))$ .

Recalled that $\varepsilon\in(0,H-1/2)$ , by Theorem 2.1 and Theorem 2.2, we have that $|X_{ij}(t)-X_{ij}(s)|\leq\xi_{ij}|t-s|^{H-\varepsilon}$ , where $\{\xi_{ij}\}_{1\leq i\leq p,1\leq j\leq N}$ are i.i.d. copies of $\xi$ with $\mathbb{E}[|\xi|^{4}]<\infty$ . Thus,

$\displaystyle\left\|\widehat{U}_{ij}^{N}(t)-\widehat{U}_{ij}^{N}(s)\right\|$	$\displaystyle\leq\left\|X_{ij}(t)-X_{ij}(s)\right\|+\left\|\mathbb{E}\left[X_{ij}(t)\right]-\mathbb{E}\left[X_{ij}(s)\right]\right\|$
	$\displaystyle\leq\left\|X_{ij}(t)-X_{ij}(s)\right\|+\mathbb{E}\left[\left\|X_{ij}(t)-X_{ij}(s)\right\|\right]$
	$\displaystyle\leq\left(\xi_{ij}+\mathbb{E}\left[\xi_{ij}\right]\right)(t-s)^{H-\varepsilon}.$	(5.3)

Hence, by (5.1), for $0\leq s<t\leq T$ ,

		$\displaystyle\quad\left\|\widehat{U}_{ik}^{N}(t)\widehat{U}_{jk}^{N}(t)-\widehat{U}_{ik}^{N}(s)\widehat{U}_{jk}^{N}(s)\right\|^{2}$
		$\displaystyle\leq 2\left\|\widehat{U}_{ik}^{N}(t)-\widehat{U}_{ik}^{N}(s)\right\|^{2}\left\|\widehat{U}_{jk}^{N}(t)\right\|^{2}+2\left\|\widehat{U}_{ik}^{N}(s)\right\|^{2}\left\|\widehat{U}_{jk}^{N}(t)-\widehat{U}_{jk}^{N}(s)\right\|^{2}$
		$\displaystyle\leq 2\left(\xi_{ik}+\mathbb{E}\left[\xi_{ik}\right]\right)^{2}\left(\left\|\widehat{U}_{jk}^{N}(0)\right\|+\left(\xi_{jk}+\mathbb{E}\left[\xi_{jk}\right]\right)T^{H-\varepsilon}\right)^{2}(t-s)^{2H-2\varepsilon}$
		$\displaystyle\quad+2\left(\xi_{jk}+\mathbb{E}\left[\xi_{jk}\right]\right)^{2}\left(\left\|\widehat{U}_{ik}^{N}(0)\right\|+\left(\xi_{ik}+\mathbb{E}\left[\xi_{ik}\right]\right)T^{H-\varepsilon}\right)^{2}(t-s)^{2H-2\varepsilon}$
		$\displaystyle=E_{H,T,\varepsilon}^{(i,j;k)}(t-s)^{2H-2\varepsilon}.$		(5.4)

Here, $E_{H,T,\varepsilon}^{(i,j;k)}$ is a positive random variable that has finite second moment which is given by

	$\displaystyle E_{H,T,\varepsilon}^{(i,j;k)}$	$\displaystyle=2\left(\xi_{ik}+\mathbb{E}\left[\xi_{ik}\right]\right)^{2}\left(\left\|\widehat{U}_{jk}^{N}(0)\right\|+\left(\xi_{jk}+\mathbb{E}\left[\xi_{jk}\right]\right)T^{H-\varepsilon}\right)^{2}$
		$\displaystyle\quad+2\left(\xi_{jk}+\mathbb{E}\left[\xi_{jk}\right]\right)^{2}\left(\left\|\widehat{U}_{ik}^{N}(0)\right\|+\left(\xi_{ik}+\mathbb{E}\left[\xi_{ik}\right]\right)T^{H-\varepsilon}\right)^{2}.$

Let, for $i\neq j$ ,

\displaystyle E_{1}=\mathbb{E}\left[E_{H,T,\varepsilon}^{(i,j;k)}\right]=4\mathbb{E}\left[\left(\xi+\mathbb{E}[\xi]\right)^{2}\right]\mathbb{E}\left[\left(\left|X_{0}-\mathbb{E}[X_{0}]\right|+\left(\xi+\mathbb{E}[\xi]\right)T^{H-\varepsilon}\right)^{2}\right],

and for $i=j$ ,

\displaystyle E_{2}=\mathbb{E}\left[E_{H,T,\varepsilon}^{(i,i;k)}\right]=4\mathbb{E}\left[\left(\xi+\mathbb{E}[\xi]\right)^{2}\left(\left|X_{0}-\mathbb{E}[X_{0}]\right|+\left(\xi+\mathbb{E}[\xi]\right)T^{H-\varepsilon}\right)^{2}\right].

Then $E_{1},E_{2}$ are two positive numbers depending only on $(H,T,\varepsilon)$ .

Without loss of generality, we assume that $\frac{p-1}{N-1}\leq c+1$ . Recall that the entries of $\widehat{U}^{N}(t)$ are independent. Using the Cauchy-Schwarz inequality twice, the mean value theorem, Lemma A.1, and (5.1), we can obtain

		$\displaystyle\quad\mathbb{E}\left[\left\|\langle f,L_{N}(t)\rangle-\langle f,L_{N}(s)\rangle\right\|^{2}\right]=\mathbb{E}\left[\left\|\dfrac{1}{p}\sum_{i=1}^{p}f(\lambda_{i}^{N}(t))-f(\lambda_{i}^{N}(s))\right\|^{2}\right]$
		$\displaystyle\leq\dfrac{1}{p}\mathbb{E}\left[\sum_{i=1}^{p}\left\|f(\lambda_{i}^{N}(t))-f(\lambda_{i}^{N}(s))\right\|^{2}\right]\leq\dfrac{\\|f^{\prime}\\|_{\infty}^{2}}{p}\mathbb{E}\left[\sum_{i=1}^{p}\left\|\lambda_{i}^{N}(t)-\lambda_{i}^{N}(s)\right\|^{2}\right]$
		$\displaystyle\leq\dfrac{\\|f^{\prime}\\|_{\infty}^{2}}{p}\mathbb{E}\left[\sum_{i,j=1}^{p}\left(U_{ij}^{N}(t)-U_{ij}^{N}(s)\right)^{2}\right]$
		$\displaystyle=\dfrac{\\|f^{\prime}\\|_{\infty}^{2}}{p}\sum_{i\not=j}^{p}\mathbb{E}\left[\left(\dfrac{1}{N}\sum_{k=1}^{N}\left[\widehat{U}_{ik}^{N}(t)\widehat{U}_{jk}^{N}(t)-\widehat{U}_{ik}^{N}(s)\widehat{U}_{jk}^{N}(s)\right]\right)^{2}\right]$
		$\displaystyle\qquad\quad+\dfrac{\\|f^{\prime}\\|_{\infty}^{2}}{p}\sum_{i=1}^{p}\mathbb{E}\left[\left(\dfrac{1}{N}\sum_{k=1}^{N}\left[\widehat{U}_{ik}^{N}(t)^{2}-\widehat{U}_{ik}^{N}(s)^{2}\right]\right)^{2}\right]$
		$\displaystyle\leq\dfrac{\\|f^{\prime}\\|_{\infty}^{2}}{pN^{2}}\sum_{i\neq j}^{p}\sum_{k=1}^{N}\mathbb{E}\left[\left(\widehat{U}_{ik}^{N}(t)\widehat{U}_{jk}^{N}(t)-\widehat{U}_{ik}^{N}(s)\widehat{U}_{jk}^{N}(s)\right)^{2}\right]$
		$\displaystyle\quad\qquad+\dfrac{\\|f^{\prime}\\|_{\infty}^{2}}{pN}\sum_{i=1}^{p}\sum_{k=1}^{N}\mathbb{E}\left[\left(\widehat{U}_{ik}^{N}(t)^{2}-\widehat{U}_{ik}^{N}(s)^{2}\right)^{2}\right]$
		$\displaystyle\leq\dfrac{\\|f^{\prime}\\|_{\infty}^{2}}{pN^{2}}\sum_{i\neq j}^{p}\sum_{k=1}^{N}\mathbb{E}\left[E_{H,T}^{(i,j;k)}\right](t-s)^{2H-2\varepsilon}+\dfrac{\\|f^{\prime}\\|_{\infty}^{2}}{pN}\sum_{i=1}^{p}\sum_{k=1}^{N}\mathbb{E}\left[E_{H,T}^{(i,i;k)}\right](t-s)^{2H-2\varepsilon}$
		$\displaystyle=\dfrac{\\|f^{\prime}\\|_{\infty}^{2}(p-1)}{N}E_{1}(t-s)^{2H-2\varepsilon}+\\|f^{\prime}\\|_{\infty}^{2}E_{2}(t-s)^{2H-2\varepsilon}$
		$\displaystyle\leq\left((c+1)E_{1}+E_{2}\right)\\|f^{\prime}\\|_{\infty}^{2}(t-s)^{2H-2\varepsilon}$		(5.5)

for any $f\in C^{1}(\mathbb{R})$ with bounded derivative. Hence, by Proposition B.3 and (5.2), we can conclude that the sequence $\{L_{N}(t),t\in[0,T]\}_{N\in\mathbb{N}}$ converges in law to $\{\mu_{t}=\mu_{MP}(c,d_{t}),t\in[0,T]\}$ . Finally, noting that the limit measure $\{\mu_{t},t\in[0,T]\}$ is deterministic, the convergence in law actually coincides with the convergence in probability.

The proof is concluded. ∎

Remark 5.1.

In contrast, the convergences of the empirical measure-valued processes obtained in Theorem 3.1 and other subsequent results in Section 3 are almost-sure convergence, which is stronger than the in-probability convergence obtained in Theorem 5.1.

In section 3, we construct a compact set in $C([0,T],\mathbf{P}(\mathbb{R}))$ and show that the sequence $\{L_{N}(t),t\in[0,T]\}$ is in that compact set almost surely. However, in the Wishart case, we are not able to get an estimation analogous to (3.1) which is the key ingredient to get the almost-sure convergence, due to the lack of the independence for the upper triangular entries. Instead, we obtain the tightness on $C([0,T],\mathbf{P}(\mathbb{R}))$ for $\{L_{N}(t),t\in[0,T]\}$ thanks to Proposition B.2, and then the convergence in law follows consequently.

Remark 5.2.

Let $\sigma(x)=1$ , $b(x)=0$ and $X_{0}=0$ , then the solution to (1.1) is the fractional Brownian motion $X_{t}=B_{t}^{H}$ . Then we have the convergence in law of the empirical spectral measures towards the scaled Marchenko-Pastur law $\mu_{MP}(c,t^{H})(dx)$ , which recovers the results obtained in [Pardo2017].

Remark 5.3.

Let $\widetilde{Y}^{N}(t)=\left(X_{ij}(t)\right)_{1\leq i\leq p,1\leq j\leq N}$ . Then under the conditions in Theorem 5.1, the sequence of empirical measures of the eigenvalues of $\frac{1}{N}\widetilde{Y}^{N}(t)\widetilde{Y}^{N}(t)^{\intercal}$ converges in probability to $\mu_{MP}(c,d_{t})(dx)$ in $C([0,T],\mathbf{P}(\mathbb{R}))$ . Indeed, by the Lidskii inequality in [Tao2012, Exercise 1.3.22 (ii)], we have

\displaystyle\left|F_{\frac{1}{N}\widetilde{Y}^{N}(t)\widetilde{Y}^{N}(t)^{\intercal}}(x)-F_{\frac{1}{N}\widehat{U}^{N}(t)\widehat{U}^{N}(t)^{\intercal}}(x)\right|\leq\dfrac{1}{N}\mathrm{rank}\left(\frac{1}{N}\widetilde{Y}^{N}(t)\widetilde{Y}^{N}(t)^{\intercal}-\frac{1}{N}\widehat{U}^{N}(t)\widehat{U}^{N}(t)^{\intercal}\right),

where $F_{A}(x)$ is the number of the eigenvalues of $A$ that are smaller than $x$ . Noting that the rank of

	$\displaystyle\quad\dfrac{1}{N}\widetilde{Y}^{N}(t)\widetilde{Y}^{N}(t)^{\intercal}-\dfrac{1}{N}\widehat{U}^{N}(t)\widehat{U}^{N}(t)^{\intercal}$
	$\displaystyle=\frac{1}{N}\left(\widehat{U}^{N}(t)+m_{t}E_{N}\right)\left(\widehat{U}^{N}(t)^{\intercal}+m_{t}E_{N}\right)-\frac{1}{N}\widehat{U}^{N}(t)\widehat{U}^{N}(t)^{\intercal}$
	$\displaystyle=\dfrac{m_{t}}{N}E_{N}\widehat{U}^{N}(t)^{\intercal}+\dfrac{m_{t}}{N}\widehat{U}^{N}(t)E_{N}+m_{t}^{2}E_{N},$

is at most $3$ for all $t\in[0,T]$ , the convergence in probability of $\{L_{N}(t)\}_{N\in\mathbb{N}}$ towards $\mu_{MP}(c,d_{t})(dx)$ implies that the empirical spectral measures of $\frac{1}{N}\widetilde{Y}^{N}(t)\widetilde{Y}^{N}(t)^{\intercal}$ converges to the same limit in probability.

Remark 5.4.

The Stieltjes transform $G_{t}(z)$ of the limiting measure $\mu_{t}$ is

\displaystyle G_{t}(z)=\int\dfrac{\mu_{t}(dx)}{z-x}=\int\dfrac{p_{MP}(c,d_{t})(x)}{z-x}dx=\int\dfrac{p_{MP}(c,1)(x)}{z-d_{t}^{2}x}dx,

where $p_{MP}(c,d_{t})(x)$ is the probability density of the Marchenko-Pastur distribution $\mu_{MP}(c,d_{t})$ given in (A.1). Assuming that the variance $d_{t}^{2}$ of the solution $X_{t}$ is continuously differentiable on $(0,T)$ , we have

$\displaystyle\partial_{t}G_{t}(z)$	$\displaystyle=\partial_{t}\int\dfrac{p_{MP}(c,1)(x)}{z-d_{t}^{2}x}dx=(d_{t}^{2})^{\prime}\int\dfrac{xp_{MP}(c,1)(x)}{(z-d_{t}^{2}x)^{2}}dx$
	$\displaystyle=\dfrac{(d_{t}^{2})^{\prime}}{d_{t}^{2}}\int\left(\dfrac{z}{(z-d_{t}^{2}x)^{2}}-\dfrac{1}{z-d_{t}^{2}x}\right)p_{MP}(c,1)(x)dx$
	$\displaystyle=-\dfrac{(d_{t}^{2})^{\prime}}{d_{t}^{2}}\left(z\partial_{z}G_{t}(z)+G_{t}(z)\right).$	(5.6)

On the other hand, [Bai and Silverstein, Lemma 3.11] and some computation yield

\displaystyle czd_{t}^{2}G_{t}(z)^{2}=G_{t}(z)\left(z-d_{t}^{2}(1-c)\right)-1.

Taking partial derivative with respect to $z$ , we have

\displaystyle cd_{t}^{2}G_{t}(z)^{2}+2czd_{t}^{2}G_{t}(z)\partial_{z}G_{t}(z)=G_{t}(z)+\partial_{z}G_{t}(z)\left(z-d_{t}^{2}(1-c)\right).

(5.7)

Therefore, by (5.4) and (5.7), we have

\displaystyle\partial_{t}G_{t}(z)=-(d_{t}^{2})^{\prime}\left(cG_{t}(z)^{2}+2czG_{t}(z)\partial_{z}G_{t}(z)+(1-c)\partial_{z}G_{t}(z)\right).

(5.8)

5.2 Complex case

Recall that $Z=(Z^{(1)},Z^{(2)})$ is the solution to (3.12). Let $\widehat{W}^{N}(t)=\left(\widehat{W}_{ij}^{N}(t)\right)_{1\leq i\leq p,1\leq j\leq N}$ be a $p\times N$ matrix with entries $\widehat{W}_{ij}^{N}(t)=Z_{ij}(t)-\mathbb{E}\left[Z_{ij}(t)\right]$ , where $Z_{ij}$ are i.i.d. copies of $Z^{(1)}+\iota Z^{(2)}$ and $p=p(N)$ is a positive integer depending on $N$ . Let

\displaystyle W^{N}(t)=\dfrac{1}{N}\widehat{W}^{N}(t)\widehat{W}^{N}(t)^{*}

(5.9)

be a $p\times p$ symmetric matrix with eigenvalue empirical measure $L_{N}(t)(dx)$ .

Theorem 5.2.

Suppose that the coefficient functions $\tilde{\sigma}$ , $\tilde{b}$ have bounded derivatives which are Hölder continuous of order greater than $1/(H-\varepsilon)-1$ . Besides, assume that one of the following conditions holds,

(a)

$\|(\tilde{\sigma}_{x},\tilde{\sigma}_{y})\|_{L^{\infty}(\mathbb{R}^{2})}+\|(\tilde{b}_{x},\tilde{b}_{y})\|_{L^{\infty}(\mathbb{R}^{2})}>0$ , $\mathbb{E}[\|Z_{0}\|^{4}]<\infty$ .
(b)

$\tilde{\sigma}$ and $\tilde{b}$ are bounded and $\mathbb{E}[\|Z_{0}\|^{2}]<\infty$ .

Moreover, suppose that there exists a positive function $\varphi(x)\in C^{1}(\mathbb{R})$ with bounded derivative, such that $\lim_{|x|\rightarrow\infty}\varphi(x)=+\infty$ and

\displaystyle\sup_{N\in\mathbb{N}}\langle\varphi,L_{N}(0)\rangle\leq C_{0},

for some positive constant $C_{0}$ almost surely. Furthermore, suppose that there exists a positive constant $c$ , such that $p/N\rightarrow c$ as $N\rightarrow\infty$ .

Then for any $T>0$ , $\mathbb{E}[\|Z_{t}\|^{2}]<\infty$ , and the sequence $\{L_{N}(t),[0,T]\}_{N\in\mathbb{N}}$ converges in probability to $\mu_{MP}(c,d_{Z}(t))(dx)$ in $C([0,T],\mathbf{P}(\mathbb{R}))$ .

Proof.

The proof is similar to the proofs of Theorem 5.1 and Theorem 3.3, which is sketched below.

From the proof of Theorem 3.3, we can obtain the finiteness of the mean $m_{Z}(t)$ and $d_{Z}^{2}(t)$ . Analogous to (5.2), by using Lemma A.5, we have the almost-sure convergence

\displaystyle L_{N}(t)(dx)\rightarrow\mu_{MP}(c,d_{Z}(t))(dx).

(5.10)

Note that the estimation (3.14) in the proof Theorem 3.3 is still valid. Similar to the estimation (5.1) and (5.1) in the proof of Theorem 5.1, we can obtain

\displaystyle\mathbb{E}\left[\left|\langle f,L_{N}(t)\rangle-\langle f,L_{N}(s)\rangle\right|^{2}\right]\leq C\|f^{\prime}\|_{\infty}^{2}(t-s)^{2H-2\varepsilon}.

Then following the argument at the end of the proof of Theorem 5.1, we can obtain the tightness of the sequence $\{L_{N}(t)\}_{N\in\mathbb{N}}$ , which implies the convergence in distribution and hence the convergence in probability, with the deterministic limit given in (5.10). ∎

Remark 5.5.

Let $\widetilde{W}^{N}(t)=\left(Z_{ij}(t)\right)_{1\leq i\leq p,1\leq j\leq N}$ . Then under the conditions in Theorem 5.2, the sequence of empirical spectral measures of $\frac{1}{N}\widetilde{W}^{N}(t)\widetilde{W}^{N}(t)^{\intercal}$ converges in probability to $\mu_{MP}(c,d_{Z})(dx)$ in $C([0,T],\mathbf{P}(\mathbb{R}))$ .

Remark 5.6.

Similar to Remark 5.4, the Stieltjes transform of the limit measure $\mu_{t}$ satisfies the differential equation (5.8) with $d_{t}$ replaced by $d_{Z}(t)$ .

Appendix A Preliminaries on (random) matrices

The following is the Hoffman-Wielandt lemma, which can be found in [Anderson2010, Lemma 2.1.19], see also [Tao2012].

Lemma A.1 (Hoffman-Wielandt).

Let $A=(A_{ij})_{1\leq i,j\leq N}$ and $B=(B_{ij})_{1\leq i,j\leq N}$ be $N\times N$ Hermitian matrices, with ordered eigenvalues $\lambda_{1}^{A}\leq\lambda_{2}^{A}\leq\ldots\leq\lambda_{N}^{A}$ and $\lambda_{1}^{B}\leq\lambda_{2}^{B}\leq\ldots\leq\lambda_{N}^{B}$ . Then

\displaystyle\sum_{i=1}^{N}\left|\lambda_{i}^{A}-\lambda_{i}^{B}\right|^{2}\leq\mathrm{Tr}\left[(A-B)(A-B)^{*}\right]=\sum_{i,j=1}^{N}\left|A_{ij}-B_{ij}\right|^{2}.

The next two lemmas are the famous Wigner semi-circle law for the real case and complex case respectively (see, e.g., [Tao2012]).

Lemma A.2.

Let $M_{N}$ be the top left $N\times N$ minors of an infinite Wigner matrix $(\xi_{ij})_{i,j\geq 1}$ , which is symmetric, the upper-triangular entries $\xi_{ij},i>j$ are i.i.d. real random variables with mean zero and unit variance, and the diagonal entries $\xi_{ii}$ are i.i.d. real variables, independent of the upper-triangular entries, with bounded mean and variance. Then the empirical spectral distributions $\mu_{M_{N}/\sqrt{N}}$ converge almost surely to the Wigner semicircular distribution

\displaystyle\mu_{sc}(dx)=\dfrac{\sqrt{4-x^{2}}}{2\pi}1_{[-2,2]}(x)dx.

Lemma A.3.

Let $M_{N}$ be the top left $N\times N$ minors of an infinite complex Wigner matrix $(\xi_{ij})_{i,j\geq 1}$ , which is Hermitian, the upper-triangular entries $\xi_{ij},i>j$ are i.i.d. complex random variables with mean zero and unit variance, and the diagonal entries $\xi_{ii}$ are i.i.d. real variables, independent of the upper-triangular entries, with bounded mean and variance. Then the conclusion of Lemma A.2 holds.

The next two lemmas concern the celebrated Marchenko-Pastur law which was introduced in [Bai2010].

Lemma A.4.

Let $X_{N}$ be the top left $p(N)\times N$ minors of an infinite random matrix, whose entries are i.i.d. real random variable with mean zero and variance $\sigma^{2}$ . Here, $p(N)$ is a positive integer such that $p(N)/N\rightarrow c\in(0,\infty)$ as $N\rightarrow\infty$ . Then the empirical distribution of the eigenvalues of the $p\times p$ matrix

\displaystyle Y^{N}=\dfrac{1}{N}X_{N}X_{N}^{\intercal}

converges weakly to the Marchenko-Pastur distribution

	$\displaystyle\mu_{MP}(c,\sigma)(dx)$	$\displaystyle=\dfrac{1}{2\pi\sigma^{2}cx}\sqrt{\left(\sigma^{2}(1+\sqrt{c})^{2}-x\right)\left(x-\sigma^{2}(1-\sqrt{c})^{2}\right)}1_{[\sigma^{2}(1-\sqrt{c})^{2},\sigma^{2}(1+\sqrt{c})^{2}]}(x)dx$
		$\displaystyle\quad+\left(1-\dfrac{1}{c}\right)\delta_{0}(x)dx1_{[c>1]}\,,$		(A.1)

almost surely, where $\delta_{0}$ is the point mass at the origin.

Lemma A.5.

Let $X_{N}$ be the top left $p(N)\times N$ minors of an infinite random matrix, whose entries are i.i.d. complex random variable with mean zero and variance $\sigma^{2}$ . Here, $p(N)$ is a positive integer such that $p(N)/N\rightarrow c\in(0,\infty)$ as $N\rightarrow\infty$ . Then the empirical distribution of the $p\times p$ matrix

\displaystyle Y^{N}=\dfrac{1}{N}X_{N}X_{N}^{*}

converges almost surely to the Marchenko-Pastur distribution $\mu_{MP}(c,\sigma)(dx)$ described in Lemma A.4.

The following result characterizes the limiting empirical spectral distribution of the symmetric random matrix with correlated entries, which is a direct corollary of [Banna2015, Theorem 3].

Lemma A.6.

Let $(\xi_{i,j})_{(i,j)\in\mathbb{Z}^{2}}$ be an array of i.i.d. real-valued random variables with finite second moment. Let $I$ be a finite subset of $\mathbb{Z}^{2}$ , $\{a_{r}:r\in I\}$ be a family of constants and

\displaystyle X_{i,j}=\sum_{r\in I}a_{r}\xi_{(i,j)+r},\quad 1\leq i\leq j.

Suppose that $\mathbb{E}[X_{0,0}]=\sum_{r\in I}a_{r}\mathbb{E}[\xi_{0,0}]=0$ . Denote $\gamma_{k,l}=\gamma_{l,k}=\mathbb{E}[X_{0,0}X_{k,l}]$ for all $k\leq l$ . Let $X^{N}=\left(X_{i,j}^{N}\right)_{1\leq i,j\leq N}$ be a symmetric matrix with entries $X_{i,j}^{N}=X_{i,j}/\sqrt{N}$ for $1\leq i\leq j\leq N$ . Then the empirical spectral measure of $X^{N}$ converges to a nonrandom probability measure $\mu_{c}$ with Stieltjes transform $S_{c}(z)=\int_{0}^{1}h(x,z)dx$ , where $h(x,z)$ is the solution to the equation

\displaystyle h(x,z)=\left(-z+\int_{0}^{1}f(x,y)h(y,z)dy\right)^{-1}~{}\text{ with }~{}f(x,y)=\sum_{k,l\in\mathbb{Z}}\gamma_{k,l}e^{-2\pi i(kx+ly)}.

Appendix B Tightness criterions for probability measures on $C([0,T],\mathbf{P}(\mathbb{R}))$

In this section, we collect some lemmas used in the proofs, and then we provide two tightness criterions for probability measures on $C([0,T],\mathbf{P}(\mathbb{R}))$ (Theorems B.1 and B.2). We also provide sufficient conditions for tightness which can be verified by computing moments (Propositions B.1, B.2 and B.3).

Note that there has been fruitful literature on tightness of probability measures on a Skorohod space $\mathcal{D}([0,T],E)$ , where $E$ is a completely regular topological space. We refer the interested reader to [Mitoma, jaku, EK, Dawson1993, Perkins, KX, Kouritzin] and the references therein. Theorem B.1 is a direct consequence of Jakubowski’s criterion [jaku] (see also e.g, [Dawson1993, Theorem 3.6.4] and [Sun2011] for the statement of the criterion), noting that $C([0,T],\mathbf{P}(\mathbb{R}))$ is a closed subset of $\mathcal{D}([0,T],\mathbf{P}(\mathbb{R}))$ . Theorem B.2 might be also well-known in the literature of tightness criterion for probability measures, but we could not find a reference addressing this explicitly. For both Theorems B.1 and B.2, we include self-contained proofs for the reader’s convenience.

Recall that $\mathbf{P}(\mathbb{R})$ is the set of probability measures on $\mathbb{R}$ endowed with its weak topology, and that $C([0,T],\mathbf{P}(\mathbb{R}))$ is the space of continuous probability-measure-valued processes, both of which are Polish spaces. Denote by $C_{0}(\mathbb{R})$ the set of continuous functions on $\mathbb{R}$ vanishing at infinity, which is also a Polish space. Also the space $\tilde{\mathbf{P}}(\mathbb{R})$ of sub-probabilities on $\mathbb{R}$ endowed with its vague topology is a Polish space (see, e.g., [Kallenberg, Theorem 4.2]), and so is $C([0,T],\tilde{\mathbf{P}}(\mathbb{R})).$

Let’s also recall some basic facts for probability measures on a Polish space $X$ (see, e.g., [Billingsley] for details). Denote by $\mathbf{P}(X)$ the set of probability measures on $(X,\mathcal{B}_{X})$ where $\mathcal{B}_{X}$ is the Borel $\sigma$ -field on the Polish space $X$ . Let $\Pi\subset\mathbf{P}(X)$ be a family of probability measures on $X$ . The family $\Pi$ is called tight if for every $\varepsilon\in(0,1)$ , there exists a compact set $K^{\varepsilon}\subset X$ such that $P(K)>1-\varepsilon$ for all $P\in\Pi$ . The family $\Pi$ is called relatively compact if every sequence of elements of $\Pi$ contains a weakly convergent subsequence. The Prokhorov’s theorem guarantees the equivalence between tightness and relatively compactness. Also note that a sequence $P_{n}\in\mathbf{P}(X)$ converges weakly to $P\in\mathbf{P}(X)$ if and only if $P_{n}$ converges to $P$ in the Polish space $\mathbf{P}(X)$ .

The following lemma (see, e.g., [Durrett2019, Theorem 3.2.14]) provides a method to obtain tightness for a set of probability measures.

Lemma B.1.

Let $\mathbb{I}$ be an index set. If there is a non-negative function $\varphi$ so that $\varphi(x)\rightarrow\infty$ as $|x|\rightarrow\infty$ and

\displaystyle\sup_{i\in\mathbb{I}}\int_{\mathbb{R}}\varphi(x)\mu_{n}(dx)<\infty,

then the family of probability measures $\{\mu_{i}\}_{i\in\mathbb{I}}$ is tight.

Based on the above tightness criterion, one can construct compact subsets of $\mathbf{P}(\mathbb{R})$ :

Lemma B.2.

A set of the form

K=\left\{\mu\in\mathbf{P}(\mathbb{R}):\int_{\mathbb{R}}\varphi(x)\mu(dx)\leq M\right\}

is compact in $\mathbf{P}(\mathbb{R})$ , where $M$ is a positive constant and $\varphi(x)$ is given in Lemma B.1.

Proof.

By Lemma B.1 and Prokhorov’s theorem (see, e.g., [Billingsley, Theorems 5.1 and 5.2]) which claims that a subset $A$ of $\mathbf{P}(\mathbb{R})$ is tight if and only if the closure of $A$ is compact, it suffices to show that $K$ is a closed set in $\mathbf{P}(\mathbb{R})$ , which is easy to verify. ∎

By the Arzela-Ascoli Theorem, we have the following lemma to construct compact sets in $C([0,T],\mathbf{P}(\mathbb{R}))$ .

Lemma B.3.

\displaystyle\mathfrak{C}=\bigcap_{n\in\mathbb{N}}\left\{g\in C([0,T],\mathbb{R}):\sup_{t,s\in[0,T],|t-s|\leq\eta_{n}}|g(t)-g(s)|\leq\varepsilon_{n},\sup_{t\in[0,T]}|g(t)|\leq M\right\},

is compact in $C([0,T],\mathbf{P}(\mathbb{R}))$ , where $M$ is a positive constant and $\{\varepsilon_{n},n\in\mathbb{N}\}$ and $\{\eta_{n},n\in\mathbb{N}\}$ are two sequences of positive numbers going to zero as $n$ goes to infinity.

The following lemma ([Anderson2010, Lemma 4.3.13]) provides an approach to construct compact subsets in $C([0,T],\mathbf{P}(\mathbb{R}))$ . It will be used in the proof of Theorem B.1.

Lemma B.4.

Let $K$ be a compact subset of $\mathbf{P}(\mathbb{R})$ , let $\{f_{i}\}_{i\in\mathbb{N}}$ be a sequence of bounded continuous functions that is dense in $C_{0}(\mathbb{R})$ , and let $\{C_{i}\}_{i\in\mathbb{N}}$ be a family of compact subsets of $C([0,T],\mathbb{R})$ . Then the set

\displaystyle\mathfrak{C}=\Big{\{}\mu\in C([0,T],\mathbf{P}(\mathbb{R})):\mu_{t}\in K,\forall t\in[0,T]\Big{\}}\cap\bigcap_{i\in\mathbb{N}}\left\{t\rightarrow\mu_{t}(f_{i})\in C_{i}\right\}

is a compact subset of $C([0,T],\mathbf{P}(\mathbb{R}))$ .

The following lemma, which constructs compacts subsets in $C([0,T],\tilde{P}(\mathbb{R}))$ , will play a critical role in the proof of Theorem B.2.

Lemma B.5.

Let $\{f_{i}\}_{i\in\mathbb{N}}$ be a countable dense subset of $C_{0}(\mathbb{R})$ , and let $\{C_{i}\}_{i\in\mathbb{N}}$ be a family of compact subsets of $C([0,T],\mathbb{R})$ . Then the set

\displaystyle\mathcal{K}=\bigcap_{i\in\mathbb{N}}\left\{t\rightarrow\mu_{t}(f_{i})\in C_{i}\right\}

is a compact subset of $C([0,T],\tilde{\mathbf{P}}(\mathbb{R}))$ .

Proof.

The proof is similar to the proof of Lemma 4.3.13 in [Anderson2010], which is provided here for the reader’s convenience.

Noting that $\mathcal{K}$ is a closed subset of $C([0,T],\tilde{\mathbf{P}}(\mathbb{R}))$ which is a Polish space, it suffices to prove that $\mathcal{K}$ is sequentially compact.

Take a sequence $\mu^{(n)}\in\mathcal{K}$ . Then the functions $t\rightarrow\mu_{t}^{(n)}(f_{i})\in C_{i}$ for $\in\mathbb{N}$ . Let $D$ be a countable dense subset of $[0,T]$ . Note that for each $t\in D$ , $\mu_{t}^{(n)}$ has a subsequence that converges vaguely, i.e., converges in the Polish space $\tilde{\mathbf{P}}(\mathbb{R})$ . Then by the diagonal procedure and the compactness of $C_{i}$ , we can find a subsequence $\mu^{\phi(n)}$ such that $t\rightarrow\mu_{t}^{\phi(n)}(f_{i})$ converges in $C_{i}$ for all $i\in\mathbb{N}$ and $\mu_{t}^{\phi(n)}$ converges in $\tilde{\mathbf{P}}(\mathbb{R})$ for all $t\in D$ , as $n$ tends to infinity. Denoting $\varphi_{i}(t):=\lim_{n\rightarrow\infty}\mu_{t}^{\phi(n)}(f_{i})$ for all $i\in\mathbb{N},t\in[0,T]$ and $\mu_{t}:=\lim_{n\rightarrow\infty}\mu_{t}^{\phi(n)}$ for all $t\in D$ , then $\varphi_{i}\in C_{i}\subset C([0,T],\mathbb{R})$ for $i\in\mathbb{N}$ and $\mu_{t}\in\tilde{\mathbf{P}}(\mathbb{R})$ for $t\in D$ . The vague convergence of the measures $\mu_{t}^{\phi(n)}$ for $t\in D$ implies that $\varphi_{i}(t)=\mu_{t}(f_{i})$ for all $i\in\mathbb{N}$ and $t\in D$ . Noting that $\{f_{i}\}_{i\in\mathbb{N}}$ is dense in $C_{0}(\mathbb{R})$ , $D$ is dense in $[0,T]$ , $\varphi_{i}(t)$ is continuous, one can extend the family $\{\mu_{t},t\in D\}$ of sub-probability measures uniquely to a sub-probability-measure-valued process $\{\nu_{t},t\in[0,T]\}\in C([0,T],\tilde{\mathbf{P}}(\mathbb{R}))$ such that $\lim_{n\rightarrow\infty}\mu^{\phi(n)}=\nu$ . This shows that $\mathcal{K}$ is sequentially compact in $C([0,T],\tilde{\mathbf{P}}(\mathbb{R}))$ , and the proof is completed. ∎

Throughout the rest of the section, let $T$ be a fixed positive number, and let $\{\mu^{(n)}\}_{n\in\mathbb{N}}:=\{\mu_{t}^{(n)},t\in[0,T]\}_{n\in\mathbb{N}}\subset C([0,T],\mathbf{P}(\mathbb{R}))$ be a sequence of continuous probability-measure-valued stochastic processes. We assume the following conditions on $\{\mu^{(n)}\}_{n\in\mathbb{N}}$ , which will be used in Theorems B.1 and B.2.

(I)

For any $\varepsilon>0$ , there exists a compact set $K^{\varepsilon}$ in $\mathbf{P}(\mathbb{R})$ , such that for all $n\in\mathbb{N}$ ,

$\mathbb{P}\left(\mu_{t}^{(n)}\in K^{\varepsilon},\forall t\in[0,T]\right)\geq 1-\varepsilon.$
(I’)

For each $t\in[0,T]$ , the family of $\mathbf{P}(\mathbb{R})$ -valued random elements $\{\mu_{t}^{(n)}:\Omega\rightarrow\mathbf{P}(\mathbb{R})\}_{n\in\mathbb{N}}$ is tight.
(II)

There exists a countable dense subset $\{f_{i}\}_{i\in\mathbb{N}}$ of $C_{0}(\mathbb{R})$ , such that for each $f_{i}$ , $\{\langle f_{i},\mu_{t}^{(n)}\rangle,t\in[0,T]\}_{n\in\mathbb{N}}$ is tight on $C([0,T],\mathbb{R})$ .

We also list some conditions that imply the above conditions.

Let $\varphi(x)$ a be nonnegative function such that $\lim\limits_{|x|\to\infty}\varphi(x)=\infty$ .

(A)

$\sup_{n\in\mathbb{N}}\mathbb{E}\left[\sup_{t\in[0,T]}\langle\varphi,\mu_{t}^{(n)}\rangle\right]<\infty.$ (B.1)
(A’)

For each $t\in[0,T]$ , the family $\{\langle\varphi,\mu_{t}^{(n)}\rangle\}_{n\in\mathbb{N}}$ of random variables is tight.
(A”)

For each $t\in[0,T]$ ,

$\sup_{n\in\mathbb{N}}\mathbb{E}\left[\left|\langle\varphi,\mu_{t}^{(n)}\rangle\right|^{\alpha}\right]<\infty,$ (B.2)

for some $\alpha>0$ ,

(B)

There exists a countable dense subset $\{f_{i}\}_{i\in\mathbb{N}}$ of $C_{0}(\mathbb{R})$ , such that there exist positive constants $\alpha$ , $\beta$ ,

\displaystyle\mathbb{E}\left[\left|\langle f,\mu_{t}^{(n)}\rangle-\langle f,\mu_{s}^{(n)}\rangle\right|^{1+\alpha}\right]\leq C_{f,T}|t-s|^{1+\beta},\quad\forall t,s\in[0,T],\forall n\in\mathbb{N},

(B.3)

for all $f\in\{f_{i}\}_{i\geq 0}$ , where $C_{f,T}$ is a constant depending only on $f$ and $T$ .

Lemma B.6.

(A) $\Rightarrow$ (I); (A”) $\Rightarrow$ (A’) $\Rightarrow(I^{\prime})$ ; (B) $\Rightarrow$ (II).

Proof.

By (B.1), let $M:=\sup_{n\in\mathbb{N}}\mathbb{E}\left[\sup_{t\in[0,T]}\langle\varphi,\mu_{t}^{(n)}\rangle\right]<\infty$ . For any $\varepsilon>0$ , choose $M_{\varepsilon}=M/\varepsilon$ . The set $K^{\varepsilon}:=\{\mu\in\mathbf{P}(\mathbb{R}):\langle\varphi,\mu\rangle\leq M_{\varepsilon}\}$ is a compact subset of $\mathbf{P}(\mathbb{R})$ by Lemma B.2. Then, by Markov inequality we have, for $n\in\mathbb{N},$

\displaystyle\mathbb{P}\left(\mu_{t}^{(n)}\notin K^{\varepsilon},\text{ for some }t\in[0,T]\right)=\mathbb{P}\left(\sup_{t\in[0,T]}\langle\varphi,\mu_{t}^{(n)}\rangle>M_{\varepsilon}\right)\leq M/M_{\varepsilon}=\varepsilon.

Thus, (A) $\Rightarrow$ (I).

(A”) $\Rightarrow$ (A’) follows directly from Lemma B.1. Now we show (A’) $\Rightarrow$ (I’). Fix an arbitrary $t\in[0,T]$ . For any $\varepsilon>0$ , due to the tightness of $\{\varphi,\mu_{t}^{(n)}\}_{n\in\mathbb{N}}$ , one can find a positive constant $N_{\varepsilon}$ such that for all $n\in\mathbb{N}$ ,

\displaystyle\mathbb{P}\left(\langle\varphi,\mu_{t}^{(n)}\rangle\leq N_{\varepsilon}\right)>1-\varepsilon.

This implies that for all $n\in\mathbb{N}$ ,

\mathbb{P}\left(\mu_{t}^{(n)}\in C^{\varepsilon}\right)>1-\varepsilon,

where $C^{\varepsilon}=\{\nu\in\mathbf{P}(\mathbb{R}):\langle\varphi,\nu\rangle\leq N_{\varepsilon}\}$ is a compact subset of $\mathbf{P}(\mathbb{R})$ by Lemma B.2. Therefore, $\{\mu_{t}^{(n)}\}_{n\in\mathbb{N}}$ is tight, and hence (A’) $\Rightarrow$ (I’).

Finally, (B) $\Rightarrow$ (II) follows directly from the Kolomogorov tightness criterion (see, e.g., [Ikeda1981, Theorem 4.2 and Theorem 4.3]). ∎

The following is Jakubowski’ tightness criterion for probability measures on $C([0,T],\mathbf{P}(\mathbb{R}))$ .

Theorem B.1.

Assume that conditions (I) and (II) are satisfied. Then the set $\{\mu_{t}^{(n)},t\in[0,T]\}_{n\in\mathbb{N}}$ induces a tight family of probability measures on $C([0,T],\mathbf{P}(\mathbb{R}))$ .

Proof.

By condition (II), for any $\varepsilon>0$ , there exist compact subsets $C_{i}^{\varepsilon}$ of $C([0,T],\mathbb{R})$ for $i\in\mathbb{N}$ , such that for each $i\in\mathbb{N}$ , for all $n\in\mathbb{N}$ ,

\displaystyle\mathbb{P}\left(\langle f_{i},\mu_{t}^{(n)}\rangle\in C_{i}^{\varepsilon}\right)\geq 1-\varepsilon/2^{i},

and hence, for all $n\in\mathbb{N}$ ,

\displaystyle\mathbb{P}\left(\bigcap_{i\in\mathbb{N}}\left\{\langle f_{i},\mu_{t}^{(n)}\rangle\in C_{i}^{\varepsilon}\right\}\right)\geq 1-\sum_{i\in\mathbb{N}}\mathbb{P}\left(\left\{\langle f,\mu_{t}^{(n)}\rangle\in C_{i}^{\varepsilon}\right\}^{\complement}\right)\geq 1-\varepsilon.

(B.4)

By Lemma B.4, the set

\displaystyle\mathfrak{C}(\varepsilon)=\Big{\{}\mu\in C([0,T],\mathbf{P}(\mathbb{R})):\mu_{t}\in K^{\varepsilon},\forall t\in[0,t]\Big{\}}\cap\bigcap_{i\geq 0}\left\{t\rightarrow\langle f_{i},\mu_{t}\rangle\in C_{i}^{\varepsilon}\right\},

is compact in $C([0,T],\mathbf{P}(\mathbb{R}))$ for any $\varepsilon>0$ . By condition (I) and (B.4), we have for all $n\in\mathbb{N}$ ,

\displaystyle\mathbb{P}\left(\mu^{(n)}\in\mathfrak{C}(\varepsilon)\right)\geq 1-2\varepsilon.

This implies the tightness of $\{\mu^{(n)}\}_{n\in\mathbb{N}}$ on $C([0,T],\mathbf{P}(\mathbb{R}))$ . The proof is concluded. ∎

Remark B.1.

Following the proof of [jaku, Theorem 3.1], one can easily show that conditions (I) and (II) are also necessary conditions for tightness of probability measures on $C([0,T],\mathbf{P}(\mathbb{R})).$

The criterion in Theorem B.1 can be verified by computing moments:

Proposition B.1.

Assume that conditions (A) and (B) are satisfied. Then the set $\{\mu_{t}^{(n)},t\in[0,T]\}_{n\in\mathbb{N}}$ induces a tight family of probability measures on $C([0,T],\mathbf{P}(\mathbb{R}))$ .

Proof.

The desired result follows directly from Theorem B.1 and Lemma B.6. ∎

In general situations, it might not be easy to check condition (I) or (A). Below, we provide another tightness criterion which weakens condition (I).

Theorem B.2.

Assume conditions (I’) and (II) are satisfied. Then the set $\{\mu_{t}^{(n)},t\in[0,T]\}_{n\in\mathbb{N}}$ induces a tight family of probability measures on $C([0,T],\mathbf{P}(\mathbb{R}))$ .

Proof.

By condition (II), we can choose the same compact subsets $C_{i}^{\varepsilon}$ of $C([0,T],\mathbb{R})$ for $i\in\mathbb{N}$ as in the proof of Theorem B.1, and hence (B.4) still holds. By Lemma B.5, the set

\displaystyle\mathcal{K}(\varepsilon)=\bigcap_{i\in\mathbb{N}}\left\{t\rightarrow\langle f_{i},\mu_{t}\rangle\in C_{i}^{\varepsilon}\right\},

is compact in $C([0,T],\tilde{\mathbf{P}}(\mathbb{R}))$ for any $\varepsilon>0$ . By (B.4), we have, for all $n\in\mathbb{N}$ ,

\displaystyle\mathbb{P}\left(\mu^{(n)}\in\mathcal{K}(\varepsilon)\right)\geq 1-\varepsilon.

This implies the tightness of $\{\mu^{(n)}\}_{n\in\mathbb{N}}$ on $C([0,T],\tilde{\mathbf{P}}(\mathbb{R}))$ . Therefore, for any subsequence of $\mu^{(n)}$ , by Prokhorov’s theorem, there exists a subsequence $\mu^{(n_{k})}$ which converge weakly to some $\nu=\{\nu_{t},t\in[0,T]\}\in C([0,T],\tilde{\mathbf{P}}(\mathbb{R}))$ which is a continuous sub-probability-measure-valued process. Thus, for each $t\in[0,T]$ , the sequence $\mu_{t}^{(n_{k})}$ (as $\tilde{\mathbf{P}}(\mathbb{R})$ -valued random elements) converges weakly to $\nu_{t}\in\tilde{\mathbf{P}}(\mathbb{R})$ . This together with the tightness of $\{\mu^{(n)}_{t}\}_{n\in\mathbb{N}}$ (as $\mathbf{P}(\mathbb{R})$ -valued random elements) in condition (I’) implies that $\nu_{t}\in\mathbf{P}(\mathbb{R})$ and hence $\nu\in C([0,T],\mathbf{P}(\mathbb{R}))$ . Therefore, Prokhorov’s theorem implies that $\{\mu^{(n)}\}_{n\in\mathbb{N}}$ is tight on $C([0,T],\mathbf{P}(\mathbb{R}))$ . The proof is concluded. ∎

Remark B.2.

Noting that condition (I) implies condition (I’), then by Remark B.1, conditions (I’) and (II) are also necessary conditions for tightness of probability measures on $C([0,T],\mathbf{P}(\mathbb{R}))$ .

Similarly, we can justify the criterion in Theorem B.2 by computing moments. The two Propositions below are direct consequences of Theorem B.2 and Lemma B.6.

Proposition B.2.

Assume that conditions (A’) and (B) are satisfied. Then the set $\{\mu_{t}^{(n)},t\in[0,T]\}_{n\in\mathbb{N}}$ induces a tight family of probability measures on $C([0,T],\mathbf{P}(\mathbb{R}))$ .

Proposition B.3.

Assume that conditions (A”) and (B) are satisfied. Then the set $\{\mu_{t}^{(n)},t\in[0,T]\}_{n\in\mathbb{N}}$ induces a tight family of probability measures on $C([0,T],\mathbf{P}(\mathbb{R}))$ .

Acknowledgment: We would like to thank Rongfeng Sun for reminding us of the Jakubowski’s tightness criterion. J. Yao is partially supported by HKSAR-RGC-Grant GRF-17307319.

	$\displaystyle\|X_{t}-X_{s}\|$	$\displaystyle\leq C(H,\varepsilon)\\|\sigma\\|_{\infty}\Big{[}\\|B^{H}\\|_{0,T;H-\varepsilon}(t-s)^{H-\varepsilon}\vee\\|B^{H}\\|_{0,T;H-\varepsilon}^{1/(H-\varepsilon)}\\|\sigma^{\prime}\\|_{\infty}^{1/(H-\varepsilon)-1}(t-s)\Big{]}$
		$\displaystyle\quad+2\\|b\\|_{\infty}(t-s).$		(2.4)

$\displaystyle\left\|D_{t-}^{1-\alpha}(B_{r}^{H}-B_{t}^{H})\right\|$	$\displaystyle=\dfrac{1}{\Gamma(\alpha)}\left\|\dfrac{B_{r}^{H}-B_{t}^{H}}{(t-r)^{1-\alpha}}+(1-\alpha)\int_{r}^{t}\dfrac{B_{r}^{H}-B_{u}^{H}}{(u-r)^{2-\alpha}}du\right\|$
	$\displaystyle\leq\dfrac{1}{\Gamma(\alpha)}\dfrac{\left\|B_{r}^{H}-B_{t}^{H}\right\|}{\|t-r\|^{1-\alpha}}+\dfrac{1-\alpha}{\Gamma(\alpha)}\int_{r}^{t}\dfrac{\left\|B_{r}^{H}-B_{u}^{H}\right\|}{\|u-r\|^{2-\alpha}}du$
	$\displaystyle\leq\dfrac{\\|B^{H}\\|_{s,t;H-\varepsilon}}{\Gamma(\alpha)}\|t-r\|^{H-\varepsilon-1+\alpha}+\dfrac{(1-\alpha)\\|B^{H}\\|_{s,t;H-\varepsilon}}{\Gamma(\alpha)}\int_{r}^{t}(u-r)^{H-\varepsilon+\alpha-2}du$
	$\displaystyle=\left(\dfrac{1}{\Gamma(\alpha)}+\dfrac{1-\alpha}{(H-\varepsilon+\alpha-1)\Gamma(\alpha)}\right)\\|B^{H}\\|_{s,t;H-\varepsilon}\|t-r\|^{H-\varepsilon-1+\alpha}\,,$	(2.6)

$\displaystyle\left\|D_{s+}^{\alpha}\sigma(X_{r})\right\|$	$\displaystyle\leq\dfrac{1}{\Gamma(1-\alpha)}\dfrac{\left\|\sigma(X_{r})\right\|}{\|r-s\|^{\alpha}}+\dfrac{\alpha}{\Gamma(1-\alpha)}\int_{s}^{r}\dfrac{\left\|\sigma(X_{r})-\sigma(X_{u})\right\|}{\|r-u\|^{\alpha+1}}du$
	$\displaystyle\leq\dfrac{\\|\sigma\\|_{\infty}}{\Gamma(1-\alpha)}\|r-s\|^{-\alpha}+\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}}{\Gamma(1-\alpha)}\int_{s}^{r}\dfrac{\left\|X_{r}-X_{u}\right\|}{\|r-u\|^{\alpha+1}}du$
	$\displaystyle\leq\dfrac{\\|\sigma\\|_{\infty}}{\Gamma(1-\alpha)}\|r-s\|^{-\alpha}+\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)}\int_{s}^{r}\dfrac{1}{\|r-u\|^{\alpha+1-(H-\varepsilon)}}du$
	$\displaystyle=\dfrac{\\|\sigma\\|_{\infty}}{\Gamma(1-\alpha)}\|r-s\|^{-\alpha}+\dfrac{\alpha\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}}{\Gamma(1-\alpha)(H-\varepsilon-\alpha)}\|r-s\|^{H-\varepsilon-\alpha}\,,$	(2.7)

$\displaystyle\left\|X_{t}-X_{s}\right\|$	$\displaystyle\leq\left\|\int_{s}^{t}\sigma(X_{r})\circ dB_{r}^{H}\right\|+\left\|\int_{s}^{t}b(X_{r})dr\right\|$
	$\displaystyle\leq C_{1}(H,\varepsilon)\\|B^{H}\\|_{0,T;H-\varepsilon}\Big{[}\\|\sigma\\|_{\infty}(t-s)^{H-\varepsilon}+\\|\sigma^{\prime}\\|_{\infty}\\|X\\|_{s,t;H-\varepsilon}(t-s)^{2H-2\varepsilon}\Big{]}$
	$\displaystyle\quad+\\|b\\|_{\infty}(t-s),$	(2.9)

	$\displaystyle\left\|X_{t}-X_{s}\right\|$	$\displaystyle\leq(t-s)^{H-\varepsilon}\\|X\\|_{s,t;H-\varepsilon}$
		$\displaystyle\leq 2C_{1}(H,\varepsilon)\\|B^{H}\\|_{0,T;H-\varepsilon}\\|\sigma\\|_{\infty}(t-s)^{H-\varepsilon}+2\\|b\\|_{\infty}(t-s).$		(2.11)

Eigenvalue distributions of high-dimensional matrix processes driven by fractional Brownian motion

Abstract

keywords:

keywords:

1 Introduction

2 Hölder continuity of the solution to the SDE (1.1)

2.1 Preliminaries on fractional calculus and fractional Brownian motion

Proposition 2.1.

Proposition 2.2.

Lemma 2.1.

2.2 Hölder continuity

Theorem 2.1.

Remark 2.1.

Proof of Theorem 2.1.

Theorem 2.2.

Proof.

Corollary 2.1.

3 High-dimensional limit for Wigner-type matrices

3.1 Relative compactness of empirical spectral measure-valued processes

Theorem 3.1.

Proof.

3.2 Limit of empirical spectral distributions

Theorem 3.2.

Proof.

Remark 3.1.

Remark 3.2.

3.3 Complex case

Theorem 3.3.

Proof.

Remark 3.3.

4 High-dimensional limit for symmetric matrices with dependent entries

Theorem 4.1.

Proof.

5 High-dimensional limit for Wishart-type matrices

5.1 Real case

Theorem 5.1.

Proof.

Remark 5.1.

Remark 5.2.

Remark 5.3.

Remark 5.4.

5.2 Complex case

Theorem 5.2.

Proof.

Remark 5.5.

Remark 5.6.

Appendix A Preliminaries on (random) matrices

Lemma A.1 (Hoffman-Wielandt).

Lemma A.2.

Lemma A.3.

Lemma A.4.

Lemma A.5.

Lemma A.6.

Appendix B Tightness criterions for probability measures on C​([0,T],𝐏​(ℝ))C([0,T],\mathbf{P}(\mathbb{R}))

Lemma B.1.

Lemma B.2.

Proof.

Lemma B.3.

Lemma B.4.

Lemma B.5.

Proof.

Lemma B.6.

Proof.

Theorem B.1.

Proof.

Remark B.1.

Proposition B.1.

Proof.

Theorem B.2.

Proof.

Remark B.2.

Proposition B.2.

Proposition B.3.

References

Appendix B Tightness criterions for probability measures on $C([0,T],\mathbf{P}(\mathbb{R}))$