The Berry-Esséen Upper Bounds of Vasicek Model Estimators

Yong Chen School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, 330022, Jiangxi, China [email protected] and Yumin Cheng School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, 330022, Jiangxi, China [email protected]

Abstract.

The Berry-Esséen upper bounds of moment estimators and least squares estimators of the mean and drift coefficients in Vasicek models driven by general Gaussian processes are studied. When studying the parameter estimation problem of Ornstein-Uhlenbeck (OU) process driven by fractional Brownian motion, the commonly used methods are mainly given by Kim and Park, they show the upper bound of Kolmogorov distance between the distribution of the ratio of two double Wiener-Itô stochastic integrals and the Normal distribution. The main innovation in this paper is extending the above ratio process, that is to say, the numerator and denominator respectively contain triple Wiener-Itô stochastic integrals at most. As far as we know, the upper bounds between the distribution of above estimators and the Normal distribution are novel.

Key words and phrases:

Vasicek model, Malliavin calculus, Central limit theorem, Berry-Esséen upper bounds

2020 Mathematics Subject Classification:

60H07,60G15,60G22

1. Introduction

Vasicek model is a type of 1-dimensional stochastic processes, it is used in various fields, such as economy, finance, environment. It was originally used to describe short-term interest rate fluctuations influenced by single market factors. Proposed by O. Vasicek [1], it is the first stochastic process model to describe the “mean reversion” characteristic of short-term interest rates. In the financial field, it can also be used as a random investment model in Wu et al.[2] and Han et al.[3].

Definition 1.

Consider the Vasicek model driven by general Gaussian process, it satisfies the following Stochastic Differential Equation (SDE):

\mathop{}\!\mathrm{d}V_{t}=k(\mu-V_{t})\mathop{}\!\mathrm{d}t+\sigma\mathop{}\!\mathrm{d}G_{t},\;\;\;t\in[0,T],

(1)

where $k,T>0,\ V_{0}=0$ and $G=\{G_{t}\}_{t\geq 0}$ is a general one-dimensional centered Gaussian process that satisfies 1.

This paper mainly focuses on the convergence rate of estimators of coefficient $k,\mu$ . Without loss of generality, we assume $\sigma=1$ , then Vasicek model can be represent by the following form:

V_{t}=\mu(1-\mathrm{e}^{-kt})+\int^{t}_{0}e^{-k(t-s)}\mathop{}\!\mathrm{d}G_{s}.

When the coefficients in the drift function is unknown, an important problem is to estimate the drift coefficients based on the observation. Based on the Brownian motion, Fergusson and Platen [4] present the maximum likelihood estimators of coefficients in Vasicek model. When the Vasicek model driven by the fractional Brownian motion, Xiao and Yu [5] consider the least squares estimators and their asymptotic behaviors. When $k>0$ , Hu and Nualart [6] study the moment estimation problem.

Since the Gaussian process $G_{t}$ mainly determines the trajectory properties of Vasicek model. Therefore, following the assumptions in Chen and Zhou [7], we make the following Hypothesis about $G_{t}$ .

Hypothesis 1 ([7] Hypothesis 1.1).

Let $\beta\in(\frac{1}{2},1)$ and $t\neq s\in[0,\infty)$ , Covariance function $R(t,s)=\mathbb{E}[G_{t}G_{s}]$ of Gaussian process $G_{t}$ satisfies the following condition:

\displaystyle\frac{\partial^{2}}{\partial t\partial s}R(t,s)

\displaystyle=C_{\beta}\left|t-s\right|^{2\beta-2}+\Psi(t,s),

(2)

where

\displaystyle\left|\Psi(t,s)\right|

\displaystyle\leq C_{\beta}^{\prime}\left|ts\right|^{\beta-1},

$\beta,\ C_{\beta}>0,C_{\beta}^{\prime}\geq 0$ are constants independent with $T$ . Besides, $R(0,t)=0$ for any $t\geq 0$ .

Remark.

The covariance functions of Gaussian processes such as fractional Brownian motion, subfractional Brownian motion and double fractional Brownian motion satisfy the above Hypothesis [7, Examples 1.5-1.8].

Assuming that there is only one trajectory $(V_{t},{t\geq 0})$ , we can construct the least squares estimators (LSE) and the moment estimators (ME) (See [8, 9, 5, 10] for more details).

Proposition 1 ([11] (4) and (5)).

The estimator of $\mu$ is the continuous-time sample mean:

\hat{\mu}=\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t.

(3)

The second moment estimator of $k$ is given by

\hat{k}=\biggl{[}\frac{\frac{1}{T}\int_{0}^{T}V_{t}^{2}\mathop{}\!\mathrm{d}t-(\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}}{C_{\beta}\Gamma(2\beta-1)}\biggr{]}^{-\frac{1}{2\beta}}.

(4)

Following from Xiao and Yu [5], we present the LSE in Vasicek model.

Proposition 2 ([11] (7) and (8)).

The LSE is motivated by the argument of minimize a quadratic function $L(k,\mu)$ of $k$ and $\mu$ :

L(k,\mu)=\int_{0}^{T}\bigl{(}\overset{\cdot}{V_{t}}-k(\mu-V_{t})\bigr{)}^{2}\mathop{}\!\mathrm{d}t,

Solving the equation, we can obtain the LSE of $k$ and $\mu$ , denoted by $\hat{k}_{LS}$ and $\hat{\mu}_{LS}$ respectively.

\hat{k}_{LS}=\dfrac{V_{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t-T\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}V_{t}}{T\int_{0}^{T}V_{t}^{2}\mathop{}\!\mathrm{d}t-(\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}},

(5)

\hat{\mu}_{LS}=\frac{V_{T}\int_{0}^{T}V_{t}^{2}\mathop{}\!\mathrm{d}t-\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}V_{t}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t}{V_{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t-T\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}V_{t}},

(6)

where the integral $\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}V_{t}$ is an Itô-Skorohod integral.

Pei et al.[11] prove the following consistencies and central limit theorems (CLT) of estimators.

Theorem 3 ([11], Theorem 1.2).

When 1 is satisfied, both ME and LSE of $\mu,k$ are strongly consistent, that is

	$\displaystyle\lim_{T\to\infty}\hat{\mu}=\mu,\;\;\;\;\;\;\lim_{T\to\infty}\hat{k}=k,\;\;\;\mathrm{a.s.};$
	$\displaystyle\lim_{T\to\infty}\hat{\mu}_{LS}=\mu,\;\;\;\lim_{T\to\infty}\hat{k}_{LS}=k,\;\;\;\mathrm{a.s.}.$

Theorem 4 ([11], Theorem 1.3).

Assume 1 is satisfied. When $G_{t}$ is self-similar and $\mathbb{E}[G_{1}^{2}]=1$ , $\hat{\mu}$ and $\hat{\mu}_{LS}$ are asymptotically normal as $T\to\infty$ , that is,

T^{1-\beta}(\hat{\mu}-\mu)\stackrel{{\scriptstyle law}}{{\longrightarrow}}\mathcal{N}(0,1/k^{2}),\;\;\;\sqrt{T}(\hat{\mu}_{LS}-\mu)\stackrel{{\scriptstyle law}}{{\longrightarrow}}\mathcal{N}(0,1/k^{2}).

When $\beta\in(\frac{1}{2},\frac{3}{4})$ ,

\sqrt{T}(\hat{k}-k)\stackrel{{\scriptstyle law}}{{\longrightarrow}}\mathcal{N}(0,k\sigma^{2}_{\beta}/4\beta^{2}),

where

\displaystyle\sigma_{\beta}^{2}=(4\beta-1)\biggl{[}1+\frac{\Gamma(3-4\beta)\Gamma(4\beta-1)}{\Gamma(2\beta)\Gamma(2-2\beta)}\biggr{]}.

(7)

Simalarly, $\sqrt{T}(\hat{k}_{LS}-k)$ is also asymptotically normal as $T\to\infty$ :

\displaystyle\sqrt{T}(\hat{k}_{LS}-k)

\displaystyle\stackrel{{\scriptstyle law}}{{\longrightarrow}}\mathcal{N}(0,k\sigma_{\beta}^{2}).

We now present the main Theorems for the whole paper, and their details are given in the following sections.

Theorem 5.

Let $Z$ be a standard Normal random variable, and $\sigma_{\beta}^{2}$ be the constant defined by (7). Assume $\beta\in(1/2,3/4)$ and 1 is satisfied. When $T$ is large enough, there exists a constant $C_{\beta,V}$ such that

	$\displaystyle\sup_{z\in\mathbb{R}}$	$\displaystyle\left\|\mathbb{P}\biggl{(}\sqrt{\frac{4\beta^{2}T}{k\sigma^{2}_{\beta}}}(\hat{k}-k)\leq z\biggr{)}-\mathbb{P}(Z\leq z)\right\|\leq\frac{C_{\beta,V}}{T^{m}},$		(8)
	$\displaystyle\sup_{z\in\mathbb{R}}$	$\displaystyle\left\|\mathbb{P}\biggl{(}\sqrt{\frac{T}{k\sigma_{\beta}^{2}}}(\hat{k}_{LS}-k)\biggr{)}-\mathbb{P}(Z\leq z)\right\|\leq\frac{C_{\beta,V}}{T^{3/4-\beta}},$		(9)

where $m=\min\{1/3,(3-4\beta)/2\}$ .

Next, we show the convergence speed of mean coefficient estimators $\hat{\mu}$ and $\hat{\mu}_{LS}$ .

Theorem 6.

Assume $\beta\in(1/2,1)$ , and $G_{t}$ is a self-similar Gaussian process satisfying 1 and $\mathbb{E}[G_{1}^{2}]=1$ . Then there exists a constant $C_{\beta,V}$ such that

	$\displaystyle\sup_{z\in\mathbb{R}}$	$\displaystyle\left\|\mathbb{P}\biggl{(}\frac{k}{T^{\beta-1}}(\hat{\mu}-\mu)\leq z\biggr{)}-\mathbb{P}(Z\leq z)\right\|\leq\frac{C_{\beta,V}}{T^{\beta/2}},$		(10)
	$\displaystyle\sup_{z\in\mathbb{R}}$	$\displaystyle\left\|\mathbb{P}\biggl{(}\frac{k}{T^{\beta-1}}(\hat{\mu}_{LS}-\mu)\biggr{)}-\mathbb{P}(Z\leq z)\right\|\leq\frac{C_{\beta,V}}{T^{(1-\beta)/2}}.$		(11)

2. Preliminary

In this section, we recall some basic facts about Malliavin calculus with respect to Gaussian process. The reader is referred to [12, 13, 14] for a more detailed explanation. Let $G=\{G_{t},t\in[0,T]\}$ be a continuous centered Gaussian process with $G_{0}=0$ and covariance function

\mathbb{E}(G_{t}G_{s})=R(s,t),\,\,s,t\in[0,T],

(12)

defined on a complete probability space $(\Omega,\mathscr{F},\mathbb{P})$ , where $\mathscr{F}$ is generated by the Gaussian family $G$ . Denote $\mathcal{E}$ as the the space of all real valued step functions on $[0,T]$ . The Hilbert space $\mathfrak{H}$ is defined as the closure of $\mathcal{E}$ endowed with the inner product:

\langle\mathbbm{1}_{[a,b)},\mathbbm{1}_{[c,d)}\rangle_{\mathfrak{H}}=\mathbb{E}((G_{b}-G_{a})(G_{d}-G_{c})).

(13)

We denote $G=\{G(h),h\in{\mathfrak{H}}\}$ as the isonormal Gaussian process on the probability space, indexed by the elements in $\mathfrak{H}$ , which satisfies the following isometry relationship:

\mathbb{E}\bigl{(}G(h)\bigr{)}=0,\,\,\mathbb{E}[G(g)G(h)]=\langle g,h\rangle_{\mathfrak{H}},\quad\forall\ g,h\in\mathfrak{H}.

(14)

The following Proposition shows the inner products representation of the Hilbert space $\mathfrak{H}$ [15].

Proposition 7 ([7] Proposition 2.1).

Denote $\mathcal{V}_{[0,T]}$ as the set of bounded variation functions on $[0,T]$ . Then $\mathcal{V}_{[0,T]}$ is dense in $\mathfrak{H}$ and

\langle f,g\rangle_{\mathfrak{H}}=\int_{[0,T]^{2}}R(t,s){v}_{f}(\mathop{}\!\mathrm{d}t){v}_{g}(\mathop{}\!\mathrm{d}s),\quad\forall\ f,g\in\mathcal{V}_{[0,T]},

where $v_{g}$ is the Lebesgue-Stieljes signed measure associated with $g^{0}$ defined as

g^{0}=\left\{\begin{array}[]{rcr}g(x),&&{\text{if}\ x\in[0,T];}\\ 0,&&\text{otherwise}.\\ \end{array}\right.

When the covariance function $R(t,s)$ satisfies 1,

\displaystyle\langle f,g\rangle_{\mathfrak{H}}=\int_{[0,T]^{2}}f(t)g(s)\frac{\partial^{2}R(t,s)}{\partial t\partial s}\mathop{}\!\mathrm{d}t\mathop{}\!\mathrm{d}s,\quad\forall\ f,g\in\mathcal{V}_{[0,T]}.

(15)

Furthermore, the norm $\left\|\cdot\right\|_{\mathfrak{H}}$ of the elements in $\mathfrak{H}$ can be induced naturally:

\left\|\phi\right\|_{\mathfrak{H}}^{2}=\int_{[0,T]^{2}}\phi(r_{1})\phi(r_{2})\frac{\partial^{2}R(r_{1},r_{2})}{\partial r_{1}\partial r_{2}}\mathop{}\!\mathrm{d}r_{1}\mathop{}\!\mathrm{d}r_{2},\;\;\;\forall\ \phi\in\mathfrak{H}.

Remark ([7] Notation 1).

Let $C_{\beta}$ and $C_{\beta}^{\prime}$ be the constants given in 1. For any $\phi(r)\in\mathcal{V}_{[0,T]}$ , we define two norms as

	$\displaystyle\left\\|\phi\right\\|_{\mathfrak{H}_{1}}^{2}$	$\displaystyle=C_{\beta}\int_{[0,T]^{2}}\phi(r_{1})\phi(r_{2})\left\|r_{1}-r_{2}\right\|^{2\beta-2}\mathop{}\!\mathrm{d}r_{1}\mathop{}\!\mathrm{d}r_{2},$
	$\displaystyle\left\\|\phi\right\\|_{\mathfrak{H}_{2}}^{2}$	$\displaystyle=C_{\beta}^{\prime}\int_{[0,T]^{2}}\left\|\phi(r_{1})\phi(r_{2})\right\|(r_{1}r_{2})^{\beta-1}\mathop{}\!\mathrm{d}r_{1}\mathop{}\!\mathrm{d}r_{2}.$

For any $\varphi(r,s)$ in $[0,T]^{2}$ , define an operator $K$ from $\mathcal{V}_{[0,T]}^{\otimes 2}$ to $\mathcal{V}_{[0,T]}$ to be

(K\varphi)(r)=\int_{0}^{T}\left|\varphi(r,u)\right|u^{\beta-1}\mathop{}\!\mathrm{d}u.

(16)

Proposition 8 ([7] Proposition 3.2).

Suppose that 1 holds, then for any $\phi(r)\in\mathcal{V}_{[0,T]}$ ,

\left|\left\|\phi\right\|_{\mathfrak{H}}^{2}-\left\|\phi\right\|_{\mathfrak{H}_{1}}^{2}\right|\leq\left\|\phi\right\|_{\mathfrak{H}_{2}}^{2},

(17)

and for any $\varphi,\psi\in(\mathcal{V}_{[0,T]})^{\odot 2}$ ,

	$\displaystyle\left\|\left\\|\phi\right\\|_{\mathfrak{H}^{\otimes{2}}}^{2}-\left\\|\phi\right\\|_{\mathfrak{H}_{1}^{\otimes{2}}}^{2}\right\|$	$\displaystyle\leq\left\\|\phi\right\\|_{\mathfrak{H}_{2}^{\otimes{2}}}^{2}+2C_{\beta}^{\prime}\left\\|K\varphi\right\\|_{\mathfrak{H}_{1}}^{2},$
	$\displaystyle\left\|\langle\varphi,\psi\rangle_{\mathfrak{H}^{\otimes{2}}}-\langle\varphi,\psi\rangle_{\mathfrak{H}_{1}^{\otimes{2}}}\right\|$	$\displaystyle\leq\left\|\langle\varphi,\psi\rangle_{\mathfrak{H}_{2}^{\otimes{2}}}\right\|+2C_{\beta}^{\prime}\left\|\langle K\varphi,K\psi\rangle_{\mathfrak{H}_{1}}\right\|.$

Let $\mathfrak{H}^{\otimes p}$ and $\mathfrak{H}^{\odot p}$ be the $p$ -th tensor product and the $p$ -th symmetric tensor product of $\mathfrak{H}$ . For every $p\geq 1$ , denote $\mathcal{H}_{p}$ as the $p$ -th Wiener chaos of $G$ . It is defined as the closed linear subspace of $L^{2}(\Omega)$ generated by $\{H_{p}(G(h)):h\in{\mathfrak{H}},\left\|h\right\|_{\mathfrak{H}}=1\}$ , where $H_{p}$ is the $p$ -th Hermite polynomial. Let $h\in\mathfrak{H}$ such that $\left\|h\right\|_{\mathfrak{H}}=1$ , then for every $p\geq 1$ and $h\in\mathfrak{H}$ ,

I_{p}(h^{\otimes p})=H_{p}\bigl{(}G(h)\bigr{)},

where $I_{p}(\cdot)$ is the $p$ -th Wiener-Itô stochastic integral.

Denote $\{e_{k},k\geq 1\}$ as a complete orthonormal system in $\mathfrak{H}$ . The $q$ -th contraction between $f\in\mathfrak{H}^{\odot m}$ and $g\in\mathfrak{H}^{\odot n}$ is an element in $\mathfrak{H}^{\otimes(m+n-2q)}$ :

\displaystyle f\otimes_{q}g=\sum_{i_{1},\cdots,i_{q}=1}^{\infty}\langle f,e_{i_{1}},\cdots,e_{i_{q}}\rangle_{\mathfrak{H}^{\otimes q}}\otimes\langle g,e_{i_{1}},\cdots,e_{i_{q}}\rangle_{\mathfrak{H}^{\otimes q}},\;\;\;q=1,\cdots,m\wedge n.

The following proposition shows the product formula for the multiple integrals.

Proposition 9 ([12] Theorem 2.7.10).

Let $f\in\mathfrak{H}^{\odot p}$ and $g\in\mathfrak{H}^{\odot q}$ be two symmetric function. Then

I_{p}(f)I_{q}(g)=\sum_{r=0}^{p\wedge q}r!\tbinom{p}{r}\tbinom{q}{r}I_{p+q-2r}(f\tilde{\otimes}_{r}g),

(18)

where $f\tilde{\otimes}_{r}g$ is the symmetrization of $f{\otimes}_{r}g$ .

We then introduce the derivative operator and the divergence operator. For these details, see sections 2.3-2.5 of [12]. Let $\mathscr{T}$ be the class of smooth random variables of the form:

F=f\bigl{(}G(\psi_{1}),G(\psi_{2}),\cdots,G(\psi_{n})\bigr{)},

where $n\geq 1$ , $f\in\mathcal{C}_{b}^{\infty}(\mathbb{R}^{n})$ which partial derivatives have at most polynomial growth, and for $i=1,\cdots,n$ , $\psi_{i}\in\mathfrak{H}$ . Then, the Malliavin derivative of $F$ (with respect to $G$ ) is the element of $L^{2}(\Omega,\mathfrak{H})$ defined by

DF=\sum_{i_{=}1}^{n}\frac{\partial f}{\partial x_{i}}\bigl{(}G(\psi_{1}),\cdots,G(\psi_{n})\bigr{)}\psi_{i}.

Given $q\in[1,\infty)$ and integer $p\geq 1$ , let $\mathbb{D}^{p,q}$ denote the closure of $\mathscr{T}$ with respect to the norm

\left\|F\right\|_{\mathbb{D}^{p,q}}=\Bigl{[}\mathbb{E}(\left|F\right|^{q})+\sum_{k=1}^{p}\mathbb{E}(\left\|D^{k}F\right\|^{q}_{\mathfrak{H}\otimes k})\Bigr{]}^{1/q}.

Denote $\delta$ (the divergence operator) as the adjoint of $D$ . The domain of $\delta$ is composed of those elements:

\left|\mathbb{E}[\langle D^{p}F,u\rangle_{\mathfrak{H}\otimes p}]\right|\leq C[\mathbb{E}(\left|F\right|^{2})]^{\frac{1}{2}},\;\;\;\forall\ F\in\mathbb{D}^{p,2},

and is denoted by $\mathrm{Dom}(\delta)$ . If $u\in\mathrm{Dom}(\delta)$ , then $\delta(u)$ is the unique element of $L^{2}(\Omega)$ characterized by the duality formula:

\mathbb{E}[F\delta(u)]=\mathbb{E}(\langle DF,u\rangle_{\mathfrak{H}}),\;\;\;\forall\ F\in\mathbb{D}^{1,2}.

We now introduce the infinitesimal generator $L$ of the Ornstein-Uhlenbeck semigroup. Let $F\in L^{2}(\Omega)$ be a square integrable random variable. Denote $\mathbb{J}_{n}:L^{2}(\Omega)\to\mathcal{H}_{n}$ as the orthogonal projection on the $n$ -th Wiener chaos $\mathcal{H}_{n}$ . The operator $L$ is defined by $LF=-\sum_{n=0}^{\infty}n\mathbb{J}_{n}F$ . The domain of $L$ is

\displaystyle\mathrm{Dom}(L):=\{F\in L^{2}(\Omega),\sum_{p=1}^{\infty}p^{2}\mathbb{E}[\mathbb{J}_{p}(F)^{2}]\leq\infty\}.

For any $F\in L^{2}(\Omega)$ , define $L^{-1}F=-\sum_{n=1}^{\infty}\frac{1}{n}\mathbb{J}_{n}F$ . $L^{-1}$ is called the Pseudo-inverse of $L$ . Note that $L^{-1}F\in\mathrm{Dom}(L)$ and $LL^{-1}F=F-\mathbb{E}(F)$ holds for any $F\in L^{2}(\Omega)$ .

The following Lemma 10 provides the Berry-Esséen upper bound on the sum of two random variables.

Lemma 10 ([16] Lemma 2).

For any variable $\xi,\eta$ and $a\in\mathbb{R}$ , the following inequality holds:

\sup_{z\in\mathbb{R}}\left|\mathbb{P}(\xi+\eta\leq z)-\Phi(z)\right|\leq\sup_{z\in\mathbb{R}}\left|\mathbb{P}(\xi\leq z)-\Phi(z)\right|+\mathbb{P}(\left|\eta\right|>a)+\frac{a}{\sqrt{2\pi}},

(19)

where $\Phi(z)$ is the standard Normal distribution function.

Using Malliavin calculus, Kim and Park [17] provide the Berry-Esséen upper bound of the quotient of two random variables.

Let $F_{T}\in\mathbb{D}^{1,2}$ be a zero-mean process, and $G_{T}\in\mathbb{D}^{1,2}$ satisfies $G_{T}>0$ a.s.. For simplicity, we define the following four functions:

	$\displaystyle\Psi_{1}(T)$	$\displaystyle=\frac{1}{(\mathbb{E}{G_{T}})^{2}}\sqrt{\mathbb{E}\Bigl{[}\Bigl{(}(\mathbb{E}G_{T})^{2}-(\langle DF_{T},-DL^{-1}F_{T}\rangle_{\mathfrak{H}})\Bigr{)}^{2}\Bigr{]}},$
	$\displaystyle\Psi_{2}(T)$	$\displaystyle=\frac{1}{(\mathbb{E}{G_{T}})^{2}}\sqrt{\mathbb{E}\Bigl{[}\langle DF_{T},-DL^{-1}(G_{T}-\mathbb{E}G_{T})\rangle_{\mathfrak{H}}^{2}\Bigr{]}},$
	$\displaystyle\Psi_{3}(T)$	$\displaystyle=\frac{1}{(\mathbb{E}{G_{T}})^{2}}\sqrt{\mathbb{E}\Bigl{[}\langle DG_{T},-DL^{-1}F_{T}\rangle_{\mathfrak{H}}^{2}\Bigr{]}},$
	$\displaystyle\Psi_{4}(T)$	$\displaystyle=\frac{1}{(\mathbb{E}{G_{T}})^{2}}\sqrt{\mathbb{E}\Bigl{[}\langle DG_{T},-DL^{-1}(G_{T}-\mathbb{E}G_{T})\rangle_{\mathfrak{H}}^{2}\Bigr{]}}.$

Theorem 11 ([17] Theorem 2 and Corollary 1).

Let $Z$ be a standard Normal variable. Assuming that for every $z\in\mathbb{R}$ , $F_{T}+zG_{T}$ has an absolutely continuous law with respect to Lebesgue measure and $\Psi_{i}(T)\to 0$ , $i=1,\cdots,4$ , as $T\to\infty$ . Then, there exists a constant $c$ such that for $T$ large enough,

\displaystyle\sup_{z\in\mathbb{R}}\left|\mathbb{P}\Bigl{(}\frac{F_{T}}{G_{T}}\leq z\Bigr{)}-\mathbb{P}(Z\leq z)\right|

\displaystyle\leq c\cdot\max_{i=1,\cdots,4}\Psi_{i}(T).

3. Berry-Esséen upper bounds of moment estimators

In this section, we will prove the Berry-Esséen upper bounds of Vasicek model moment estimators $\hat{\mu}$ and $\hat{k}$ . For the convenience of the following discussion, we first define $A(z)$ :

\displaystyle A(z):=\mathbb{P}\biggl{(}\frac{k}{T^{\beta-1}}(\hat{\mu}-\mu)\leq z\biggr{)}-\mathbb{P}(Z\leq z),

(20)

where $Z\sim\mathcal{N}(0,1)$ is a standard Normal variable. Next, we introduce the CLT of $\hat{\mu}$ .

Theorem 12 ([11] Proposition 4.19).

Assume $\beta\in(1/2,1)$ , and $G_{t}$ is a self-similar Gaussian process satisfying 1 and $\mathbb{E}[G_{1}^{2}]=1$ . Then $T^{1-\beta}(\hat{\mu}-\mu)$ is asymptotically normal as $T\to\infty$ :

\displaystyle T^{1-\beta}(\hat{\mu}-\mu)=\frac{\mu}{k}\frac{\mathrm{e}^{-kT}-1}{T^{\beta}}+\frac{1}{k}\frac{G_{T}-Z_{T}}{T^{\beta}}\stackrel{{\scriptstyle law}}{{\longrightarrow}}\mathcal{N}(0,1/k^{2}),

(21)

where

Z_{T}=I_{1}\bigl{(}\mathrm{e}^{-k(T-s)}\mathbbm{1}_{[0,T]}(s)\bigr{)}

is stochastic integral with respect to $G_{t}$ .

Following from the above Theorem, we can obtain the expanded form of (20):

	$\displaystyle A(z)$	$\displaystyle=\mathbb{P}\Bigl{(}kT^{1-\beta}(\hat{\mu}-\mu)\leq z\Bigr{)}-\mathbb{P}(Z\leq z)$
		$\displaystyle=\mathbb{P}\biggl{(}\frac{G_{T}-Z_{T}+\mu(\mathrm{e}^{-kT}-1)}{T^{\beta}}\leq z\biggr{)}-\mathbb{P}(Z\leq z).$

Then, we can prove the convergence speed of $\hat{\mu}$ .

Proof of formula (10).

Let $a=T^{-\beta/2}$ , According to Lemma 10, we have

	$\displaystyle\sup_{z\in\mathbb{R}}\left\|A(z)\right\|$	$\displaystyle\leq\sup_{z\in\mathbb{R}}\left\|\mathbb{P}\Bigl{(}\frac{G_{T}}{T^{\beta}}\leq z\Bigr{)}-\mathbb{P}(Z\leq z)\right\|$
		$\displaystyle\ \ \ +\mathbb{P}\biggl{(}\left\|\frac{-Z_{T}+\mu(\mathrm{e}^{-kT}-1)}{T^{\beta}}\right\|>T^{-\frac{\beta}{2}}\biggr{)}+\frac{T^{-\frac{\beta}{2}}}{\sqrt{2\pi}}.$

Since $G_{T}$ is self-similar, $(G_{T}/{T^{\beta}})$ is standard Normal variable,

\displaystyle\sup_{z\in\mathbb{R}}\left|\mathbb{P}\Bigl{(}\frac{G_{T}}{T^{\beta}}\leq z\Bigr{)}-\mathbb{P}(Z\leq z)\right|=0.

Following from Chebyshev inequality, we can obtain

\displaystyle\mathbb{P}\biggl{(}\left|\frac{-Z_{T}+\mu(\mathrm{e}^{-kT}-1)}{T^{\beta}}\right|>T^{-\frac{\beta}{2}}\biggr{)}=\mathbb{P}\bigl{(}\left|Z_{T}-\mu(\mathrm{e}^{-kT}-1)\right|>T^{\frac{\beta}{2}}\bigr{)}\leq\frac{C_{1}}{T^{\beta}},

where

\displaystyle C_{1}=\mathbb{E}\bigl{(}\left|Z_{T}-\mu(\mathrm{e}^{-kT}-1)\right|^{2}\bigr{)}.

The Proposition 3.10 of [11] ensures that $C_{1}$ is bounded. Combining the above results, we have

\displaystyle\sup_{z\in\mathbb{R}}\left|A(z)\right|\leq\frac{1}{\sqrt{2\pi}T^{\beta/2}}+\frac{C_{1}}{T^{\beta}}.

(22)

When $T$ is sufficiently large, there exists the constant $C_{\beta,V}$ such that the formula (10) holds. ∎

Similarly, we review the central limit theorem of $\hat{k}$ .

Theorem 13 ([11] Proposition 4.18).

Assume $\beta\in(1/2,3/4)$ and $G_{t}$ is a Gaussian process satisfying 1. Then $\sqrt{T}(\hat{k}-k)$ is asymptotically normal as $T\to\infty$ :

\displaystyle\sqrt{T}(\hat{k}-k)=\sqrt{T}\left(\left[\frac{\frac{1}{T}\int_{0}^{T}V_{t}^{2}dt-(\frac{1}{T}\int_{0}^{T}V_{t}dt)^{2}}{C_{\beta}\Gamma(2\beta-1)}\right]^{-\frac{1}{2\beta}}-k\right)\stackrel{{\scriptstyle law}}{{\longrightarrow}}\mathcal{N}(0,k\sigma^{2}_{\beta}/4\beta^{2}).

The following Lemma shows the upper bound of the expectation of $(\int_{0}^{T}\mathrm{e}^{-ks}\mathop{}\!\mathrm{d}G_{s})^{2}$ .

Lemma 14.

Let $M_{T}$ be the process defined by

M_{T}=I_{1}\bigl{(}\mathrm{e}^{-ku}\mathbbm{1}_{[0,T]}(u)\bigr{)}=\int_{0}^{T}\mathrm{e}^{-ks}\mathop{}\!\mathrm{d}G_{s}.

When $\beta\in(1/2,1)$ , there exists constant $C$ independent of $T$ such that

\mathbb{E}(M^{2}_{T})\leq C.

(23)

Proof.

According to (14) and (17), we can obtain

\displaystyle\mathbb{E}[\left|M_{T}\right|^{2}]=\left\|m_{T}\right\|_{\mathfrak{H}}^{2}\leq\left\|m_{T}\right\|^{2}_{\mathfrak{H}_{1}}+\left\|m_{T}\right\|^{2}_{\mathfrak{H}_{2}},

where $m_{T}(u)=\mathrm{e}^{-ku}\mathbbm{1}_{[0,T]}(u)$ .

It is easy to see that

	$\displaystyle\left\\|m_{T}\right\\|^{2}_{\mathfrak{H}_{1}}$	$\displaystyle=2C_{\beta}\int_{0\leq u\leq v\leq T}\mathrm{e}^{-k(u+v)}\left\|u-v\right\|^{2\beta-2}\mathop{}\!\mathrm{d}u\mathop{}\!\mathrm{d}v$
		$\displaystyle\leq C\int_{0}^{T}(\int_{0}^{T}\mathrm{e}^{-k(2x+y)}x^{2\beta-2}\mathop{}\!\mathrm{d}x)\mathop{}\!\mathrm{d}y\leq C^{\prime},$		(24)

where $C^{\prime}$ is a constant. Also, we have

\displaystyle\left\|m_{T}\right\|^{2}_{\mathfrak{H}_{2}}

\displaystyle=C^{\prime}_{\beta}\int_{0}^{T}\mathrm{e}^{-ku}u^{\beta-1}\mathop{}\!\mathrm{d}u\int_{0}^{T}\mathrm{e}^{-kv}v^{\beta-1}\mathop{}\!\mathrm{d}v\leq C^{\prime\prime}.

(25)

Combining the above two formulas, we obtain (23). ∎

Denote $B(z)$ as

\displaystyle B(z):

\displaystyle=\mathbb{P}\biggl{(}\sqrt{\frac{4\beta^{2}T}{k\sigma^{2}_{\beta}}}(\hat{k}-k)\leq z\biggr{)}-\mathbb{P}(Z\leq z).

Then we can obtain the Berry-Esséen upper bound of ME $\hat{k}$ .

Proof of formula (8).

According to [11] Proposition 4.18, we have

	$\displaystyle B(z)$	$\displaystyle=\mathbb{P}\biggl{(}\sqrt{\frac{4\beta^{2}T}{k\sigma^{2}_{\beta}}}(\hat{k}-k)\leq z\biggr{)}-\mathbb{P}(Z\leq z)$
		$\displaystyle=\mathbb{P}\biggl{(}\hat{k}-k\leq\sqrt{\frac{k\sigma^{2}_{\beta}}{4\beta^{2}T}}z\biggr{)}-\mathbb{P}(Z\leq z)$
		$\displaystyle=\mathbb{P}\biggl{(}\frac{\frac{1}{T}\int_{0}^{T}V^{2}_{t}\mathop{}\!\mathrm{d}t-\bigl{(}\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t\bigr{)}^{2}}{C_{\beta}\Gamma(2\beta-1)}\geq\biggl{(}\sqrt{\frac{k\sigma^{2}_{\beta}}{4\beta^{2}T}}z+k\biggr{)}^{-2\beta}\biggr{)}-\mathbb{P}(Z\leq z)$
		$\displaystyle=\mathbb{P}\biggl{(}\frac{1}{T}\bigl{(}\int_{0}^{T}V^{2}_{t}\mathop{}\!\mathrm{d}t-(\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}\bigr{)}-\alpha$
		$\displaystyle\;\;\;\;\;\;\;\;\;\geq C_{\beta}\Gamma(2\beta-1)\biggl{[}\biggl{(}\sqrt{\frac{k\sigma^{2}_{\beta}}{4\beta^{2}T}}z+k\biggr{)}^{-2\beta}-k^{-2\beta}\biggr{]}\biggr{)}-\mathbb{P}(Z\leq z)$
		$\displaystyle=\mathbb{P}\biggl{(}\frac{1}{T}\bigl{(}\int_{0}^{T}V^{2}_{t}\mathop{}\!\mathrm{d}t-(\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}\bigr{)}-\alpha\geq\alpha\biggl{[}\biggl{(}1+\frac{z\sigma_{\beta}}{2\beta\sqrt{kT}}\biggr{)}^{-2\beta}-1\biggr{]}\biggr{)}$
		$\displaystyle\ \ \ -\mathbb{P}(Z\leq z).$

We denote $\overline{\Phi}(z)$ as the tail probability $1-\mathbb{P}(Z\leq z)$ and

\nu=\sqrt{\frac{kT}{\sigma_{\beta}^{2}}}\biggl{[}\biggl{(}1+\frac{z\sigma_{\beta}}{2\beta\sqrt{kT}}\biggr{)}^{-2\beta}-1\biggr{]}.

Then we can obtain

$\displaystyle\left\|B(z)\right\|$	$\displaystyle=\left\|\mathbb{P}\biggl{(}\sqrt{\frac{kT}{\alpha^{2}\sigma_{\beta}^{2}}}\biggl{[}\frac{1}{T}\bigl{(}\int_{0}^{T}V^{2}_{t}\mathop{}\!\mathrm{d}t-(\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}\bigr{)}-\alpha\biggr{]}\geq\nu\biggr{)}-\mathbb{P}(Z\leq z)\right\|$
	$\displaystyle\leq\left\|\mathbb{P}\biggl{(}\sqrt{\frac{kT}{\alpha^{2}\sigma_{\beta}^{2}}}\biggl{[}\frac{1}{T}\bigl{(}\int_{0}^{T}V^{2}_{t}\mathop{}\!\mathrm{d}t-(\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}\bigr{)}-\alpha\biggr{]}\geq\nu\biggr{)}-\overline{\Phi}(\nu)\right\|$
	$\displaystyle\ \ \ +\left\|\overline{\Phi}(\nu)-\mathbb{P}(Z\leq z)\right\|$
	$\displaystyle=\left\|D(\nu)\right\|+\left\|\overline{\Phi}(\nu)-\mathbb{P}(Z\leq z)\right\|.$	(26)

Denote $D(v)$ as

\displaystyle D(\nu):

\displaystyle=\left|\mathbb{P}\biggl{(}\sqrt{\frac{kT}{\alpha^{2}\sigma_{\beta}^{2}}}\biggl{[}\frac{1}{T}\bigl{(}\int_{0}^{T}V^{2}_{t}\mathop{}\!\mathrm{d}t-(\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}\bigr{)}-\alpha\biggr{]}\geq\nu\biggr{)}-\overline{\Phi}(\nu)\right|,

where $\nu\in\mathbb{R}$ , $\alpha:=C_{\beta}\Gamma(2\beta-1)k^{-2\beta}$ . The Lemma 5.4 of [7] ensures that

\left|\overline{\Phi}(\nu)-\mathbb{P}(Z\leq z)\right|\leq\frac{C}{\sqrt{T}}.

Combining with Lemma 15, we obtain the desired result. ∎

The following Lemma provides the upper bound of $D(\nu)$ .

Lemma 15.

When $T$ is large enough, there exists constant $C_{\beta,V}^{\prime}$ such that

\displaystyle\sup_{\nu\in\mathbb{R}}\left|D(\nu)\right|\leq\frac{C_{\beta,V}^{\prime}}{T^{m}},

where $m=\min\ \{1/3,(3-4\beta)/2\}$ .

Proof.

Since the Normal distribution is symmetric, we have

	$\displaystyle D(\nu)$	$\displaystyle=\left\|\mathbb{P}\biggl{(}\sqrt{\frac{kT}{\alpha^{2}\sigma_{\beta}^{2}}}\biggl{[}\frac{1}{T}\bigl{(}\int_{0}^{T}V^{2}_{t}\mathop{}\!\mathrm{d}t-(\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}\bigr{)}-\alpha\biggr{]}\geq\nu\biggr{)}-\overline{\Phi}(\nu)\right\|$
		$\displaystyle=\left\|\mathbb{P}\biggl{(}\sqrt{\frac{kT}{\alpha^{2}\sigma_{\beta}^{2}}}\biggl{[}\frac{1}{T}\bigl{(}\int_{0}^{T}V^{2}_{t}\mathop{}\!\mathrm{d}t-(\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}\bigr{)}-\alpha\biggr{]}\leq\nu\biggr{)}-\Phi(\nu)\right\|.$

Consider the following processes:

	$\displaystyle X_{t}$	$\displaystyle=\int_{0}^{t}\mathrm{e}^{-k(t-s)}\mathop{}\!\mathrm{d}G_{s},\;\;\;I_{5}=\frac{1}{T}\biggl{(}\int_{0}^{T}X_{t}^{2}\mathop{}\!\mathrm{d}t\biggr{)},$		(27)
	$\displaystyle F_{T}$	$\displaystyle=\int_{0}^{T}\int_{0}^{t}\mathrm{e}^{-k(t-s)}\mathop{}\!\mathrm{d}G_{s}\mathop{}\!\mathrm{d}t,$

where $X_{t}$ is an OU process driven by $G_{t}$ . According to [11] formula (63), we can obtain

\displaystyle\frac{1}{T}\int_{0}^{T}V_{t}^{2}\mathop{}\!\mathrm{d}t-\biggl{(}\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t\biggr{)}^{2}-\alpha=\frac{1}{T}\biggl{(}\int_{0}^{T}X_{t}^{2}\mathop{}\!\mathrm{d}t\biggr{)}-\alpha+K_{T}+I_{4},

where

	$\displaystyle K_{T}$	$\displaystyle=\frac{\mu^{2}}{2k}\cdot\frac{1-\mathrm{e}^{-2kT}}{T}-\frac{\mu^{2}(\mathrm{e}^{-kT}-1)^{2}}{k^{2}T^{2}},$
	$\displaystyle I_{4}$	$\displaystyle=\frac{\mu}{k}\biggl{[}\frac{Z_{T}}{T}\mathrm{e}^{-kT}-\frac{M_{T}}{T}+2(1-\mathrm{e}^{-kT})\frac{F_{T}}{T^{2}}\biggr{]}-\frac{F_{T}^{2}}{T^{2}}.$

Let $a=T^{-1/3}$ . Lemma 10 ensures that

\begin{split}\sup_{\nu\in\mathbb{R}}\left|D(\nu)\right|&\leq\sup_{\nu\in\mathbb{R}}\left|{\mathbb{P}\biggl{(}\sqrt{\frac{kT}{\alpha^{2}\sigma_{\beta}^{2}}}\bigl{(}I_{5}-\alpha+K_{T}\bigr{)}\leq\nu\biggr{)}}-\mathbb{P}(Z\leq\nu)\right|\\ &\ \ \ +\mathbb{P}\biggl{(}\left|\sqrt{\frac{kT}{\alpha^{2}\sigma_{\beta}^{2}}}I_{4}\right|>T^{-1/3}\biggr{)}+\frac{T^{-1/3}}{\sqrt{2\pi}}.\end{split}

(28)

According to [7] Theorem 1.4, we have

\sup_{\nu\in\mathbb{R}}\left|{\mathbb{P}\biggl{(}\sqrt{\frac{kT}{\alpha^{2}\sigma_{\beta}^{2}}}\bigl{(}I_{5}-\alpha+K_{T}\bigr{)}\leq\nu\biggr{)}}-\mathbb{P}(Z\leq\nu)\right|\leq\frac{C_{k,\beta}}{T^{\frac{3-4\beta}{2}}},

(29)

where $C_{k,\beta}$ independent of $T$ is a constant. Denote $K_{1}=4\sqrt{\frac{k}{\alpha^{2}\sigma_{\beta}^{2}}}$ . We then consider the second term of right side of (28).

	$\displaystyle\mathbb{P}\biggl{(}\left\|\sqrt{\frac{kT}{\alpha^{2}\sigma_{\beta}^{2}}}I_{4}\right\|>T^{-1/3}\biggr{)}$	$\displaystyle\leq\mathbb{P}\biggl{(}\left\|K_{1}\frac{\mu\cdot\mathrm{e}^{-kT}\cdot Z_{T}}{k}\right\|>T^{1/6}\biggr{)}$
		$\displaystyle\ \ \ +\mathbb{P}\biggl{(}\left\|K_{1}\frac{\mu\cdot M_{T}}{k}\right\|>T^{1/6}\biggr{)}$
		$\displaystyle\ \ \ +\mathbb{P}\biggl{(}\left\|K_{1}\frac{\mu\cdot(2-2\mathrm{e}^{-kT})\cdot F_{T}}{k}\right\|>T^{7/6}\biggr{)}$
		$\displaystyle\ \ \ +\mathbb{P}\biggl{(}\left\|K_{1}\cdot F^{2}_{T}\right\|>T^{7/6}\biggr{)}.$

Combining the Chebyshev inequality, Lemma 14 and the Proposition 3.10 of [11], we can obtain

	$\displaystyle\mathbb{P}\biggl{(}\left\|K_{1}\frac{\mu\cdot\mathrm{e}^{-kT}\cdot Z_{T}}{k}\right\|>T^{1/6}\biggr{)}$	$\displaystyle\leq\frac{C^{\prime}_{1}\mathbb{E}(Z_{T}^{2})}{T^{1/3}}\leq\frac{C_{1}}{T^{1/3}},$
	$\displaystyle\mathbb{P}\biggl{(}\left\|K_{1}\frac{\mu\cdot M_{T}}{k}\right\|>T^{1/6}\biggr{)}$	$\displaystyle\leq\frac{C^{\prime}_{2}\mathbb{E}(M_{T}^{2})}{T^{1/3}}\leq\frac{C_{2}}{T^{1/3}},$
	$\displaystyle\mathbb{P}\biggl{(}\left\|K_{1}\frac{\mu\cdot(2-2\mathrm{e}^{-kT})\cdot F_{T}}{k}\right\|>T^{13/6}\biggr{)}$	$\displaystyle\leq\frac{C^{\prime}_{3}\mathbb{E}(F_{T}^{2})}{T^{13/3}}\leq\frac{C_{3}}{T^{13/3-2\beta}},$
	$\displaystyle\mathbb{P}\Bigl{(}\left\|K_{1}\cdot F^{2}_{T}\right\|>T^{13/6}\Bigr{)}$	$\displaystyle\leq\frac{C^{\prime}_{4}\mathbb{E}(F_{T}^{2})}{T^{13/6}}\leq\frac{C_{4}}{T^{13/6-2\beta}}.$

Then we have

\mathbb{P}\biggl{(}\left|\sqrt{\frac{kT}{a^{2}\sigma_{\beta}^{2}}}I_{4}\right|>T^{-1/3}\biggr{)}+\frac{T^{-1/3}}{\sqrt{2\pi}}\leq\frac{C_{k,\beta}^{\prime}}{T^{1/3}},

(30)

where $C_{k,\beta}^{\prime}$ is a constant. Combining formulas (29) and (30), we obtain the desired result. ∎

4. Berry-Esséen upper bounds of least squares estimators

For the convenience of following proof, we introduce some variables:

	$\displaystyle a_{T}$	$\displaystyle=1-\mathrm{e}^{-kT},\;\;\;b_{T}=\frac{1}{T}\int_{0}^{T}\left\\|\mathrm{e}^{-k(t-\cdot)}\mathbbm{1}_{[0,t]}(\cdot)\right\\|_{\mathfrak{H}}^{2}\mathop{}\!\mathrm{d}t\to C_{\beta}\Gamma(2\beta-1)k^{-2\beta},$
	$\displaystyle c_{T}$	$\displaystyle=\int_{0}^{T}\mu^{2}(1-\mathrm{e}^{-kt})^{2}\mathop{}\!\mathrm{d}t=\mu^{2}(T+\frac{2}{k}(\mathrm{e}^{-kT}-1)+\frac{1}{2k}(1-\mathrm{e}^{-2kT})),$
	$\displaystyle d_{T}$	$\displaystyle=\int_{0}^{T}(1-\mathrm{e}^{-kt})\mathop{}\!\mathrm{d}t=T+\frac{1}{k}(\mathrm{e}^{-kT}-1).$

The Proposition 3.10 of [11] and Proposition 9 ensure that

	$\displaystyle e_{T}$	$\displaystyle=\left\\|l_{T}\otimes_{1}l_{T}\right\\|_{\mathfrak{H}}^{2}\leq CT^{2\beta},$
	$\displaystyle q_{T}$	$\displaystyle=\left\\|l_{T}\otimes_{1}k_{T}\right\\|_{\mathfrak{H}}^{2}\leq\left\\|k_{T}\otimes_{1}\frac{1}{k}\right\\|_{\mathfrak{H}}^{2}\leq CT^{2\beta},$

where $C$ is a constant independent of $T$ . Also, we show $l_{T}$ and other functions:

$\displaystyle f_{T}(t,s)$	$\displaystyle=\mathrm{e}^{-k\left\|t-s\right\|}\mathbbm{1}_{[0,T]^{2}}(t,s),\;\;$	$\displaystyle h_{T}(t,s)=\mathrm{e}^{-k(T-t)-k(T-s)}\mathbbm{1}_{[0,T]^{2}}(t,s),$
$\displaystyle g_{T}(t,s)$	$\displaystyle=\frac{1}{2kT}f_{T}-h_{T},$
$\displaystyle k_{T}(s)$	$\displaystyle=\mathrm{e}^{-k(T-s)}\mathbbm{1}_{[0,T]}(s),$	$\displaystyle l_{T}(s)=\frac{1}{k}(1-\mathrm{e}^{-k(T-s)})\mathbbm{1}_{[0,T]}(s),$
$\displaystyle m_{T}(s)$	$\displaystyle=\mathrm{e}^{-ks}\mathbbm{1}_{[0,T]}(s),$	$\displaystyle n_{T}(s)=\frac{1}{2k}(\mathrm{e}^{-k(2T-s)}-1)\mathbbm{1}_{[0,T]}(s).$

Furthermore, we denote $I_{1}(f_{T})$ as $I_{1}(f_{T}(t,\cdot)\mathbbm{1}_{[0,T]}(\cdot))$ .

We now extent the Corollary 1 of [17].

Theorem 16.

Let $F_{T}/H_{T}$ be a zero-mean ratio process that contains at most triple Wiener-Itô stochastic integrals, where $F_{T}=I_{1}(f_{1,T})+I_{2}(f_{2,T})+I_{3}(f_{3,T})$ , $H_{T}=E_{T}+I_{1}(h_{1,T})+I_{2}(h_{2,T})+I_{3}(h_{3,T})$ and $E_{T}$ is a positive function of $T$ converging to a constant $\alpha$ . Suppose that $\Psi_{i}(T)\to 0$ , $i=1,\cdots,4$ , as $T\to\infty$ . Then, there exists constant $C$ such that when $T$ is large enough,

\begin{split}\sup_{z\in\mathbb{R}}\left|\mathbb{P}\Bigl{(}\frac{F_{T}}{H_{T}}\leq z\Bigr{)}-\mathbb{P}(Z\leq z)\right|&\leq C\cdot\max_{\begin{subarray}{c}p,q=f,h\\ m,n=1,2,3\\ i=1,\cdots,(m\wedge n)\end{subarray}}\Bigl{(}\left\|p_{m,T}\tilde{\otimes}_{i}q_{n,T}\right\|_{\mathfrak{H}^{\otimes(m+n-2i)}},\\ &\;\;\;\left\langle p_{m,T},q_{m,T}\right\rangle_{\mathfrak{H}^{\otimes m}},(E_{T}^{2}-\sum_{j=1}^{m}j!\left\|f_{j,T}\right\|_{\mathfrak{H}^{\otimes j}}^{2})\Bigr{)}.\end{split}

(31)

Proof.

We first consider $\Psi_{4}(T)=\frac{1}{(\mathbb{E}H_{T})^{2}}\sqrt{\mathbb{E}[\left\langle DH_{T},-DL^{-1}(H_{T}-\mathbb{E}H_{T})\right\rangle^{2}_{\mathfrak{H}}]}$ . It is easy to see that $(\mathbb{E}H_{T})^{2}\to E_{T}^{2}\to\alpha^{2}\;\mathrm{a.s.}$ . Then we deal with $\sqrt{\mathbb{E}[\left\langle DH_{T},-DL^{-1}(H_{T}-\mathbb{E}H_{T})\right\rangle^{2}_{\mathfrak{H}}]}$ . Denote $\left\langle DH_{T},-DL^{-1}(H_{T}-\mathbb{E}H_{T})\right\rangle_{\mathfrak{H}}=\psi_{4}$ , we have

$\displaystyle\psi_{4}$	$\displaystyle=\left\langle h_{1,T}+2I_{1}(h_{2,T})+3I_{2}(h_{3,T}),h_{1,T}+I_{1}(h_{2,T})+I_{2}(h_{3,T})\right\rangle_{\mathfrak{H}}$
	$\displaystyle=\left\\|h_{1,T}\right\\|_{\mathfrak{H}}^{2}+3h_{1,T}I_{1}(h_{2,T})+4h_{1,T}I_{2}(h_{3,T})$
	$\displaystyle\;\;\;+2\left\\|h_{2,T}\right\\|_{\mathfrak{H}^{\otimes 2}}^{2}+2I_{2}(h_{2,T}\tilde{\otimes}_{1}h_{2,T})$	(32)
	$\displaystyle\;\;\;+5I_{3}(h_{2,T}\tilde{\otimes}_{1}h_{3,T})+10I_{1}(h_{2,T}\tilde{\otimes}_{2}h_{3,T})$
	$\displaystyle\;\;\;+6\left\\|h_{3,T}\right\\|_{\mathfrak{H}^{\otimes 2}}^{2}+3I_{4}(h_{3,T}\tilde{\otimes}_{1}h_{3,T})+12I_{2}(h_{3,T}\tilde{\otimes}_{2}h_{3,T}).$

Following from the the orthogonality property of multiple integrals, we can obtain

\displaystyle\Psi_{4}(T)

\displaystyle\leq C_{1}\cdot\max_{\begin{subarray}{c}m,n=1,2,3\\ i=1,\cdots,(m\wedge n)\end{subarray}}\Bigl{(}\left\|h_{m,T}\tilde{\otimes}_{i}h_{n,T}\right\|_{\mathfrak{H}^{\otimes(m+n-2i)}},\left\|h_{m,T}\right\|_{\mathfrak{H}^{\otimes m}}^{2}\Bigr{)},

(33)

where $C_{1}$ is a constant independent of $T$ .

We next consider $\Psi_{2}(T)$ and $\Psi_{3}(T)$ . Denote $\left\langle DF_{T},-DL^{-1}(H_{T}-\mathbb{E}H_{T})\right\rangle_{\mathfrak{H}}=\psi_{2}$ , we have

	$\displaystyle\psi_{2}$	$\displaystyle=\left\langle f_{1,T}+2I_{1}(f_{2,T})+3I_{2}(f_{3,T}),h_{1,T}+I_{1}(h_{2,T})+I_{2}(h_{3,T})\right\rangle_{\mathfrak{H}}$
		$\displaystyle=f_{1,T}h_{1,T}+f_{1,T}I_{1}(h_{2,T})+f_{1,T}I_{2}(h_{3,T})$
		$\displaystyle\;\;\;+2h_{1,T}I_{1}(f_{2,T})+3h_{1,T}I_{2}(f_{3,T})$
		$\displaystyle\;\;\;+2\left\langle f_{2,T},h_{2,T}\right\rangle_{\mathfrak{H}^{\otimes 2}}+2I_{2}(f_{2,T}\tilde{\otimes}_{1}h_{2,T})$
		$\displaystyle\;\;\;+2I_{3}(f_{2,T}\tilde{\otimes}_{1}h_{3,T})+4I_{1}(f_{2,T}\tilde{\otimes}_{2}h_{3,T})$
		$\displaystyle\;\;\;+3I_{3}(f_{3,T}\tilde{\otimes}_{1}h_{2,T})+6I_{1}(f_{3,T}\tilde{\otimes}_{2}h_{2,T})$
		$\displaystyle\;\;\;+6\left\langle f_{3,T},h_{3,T}\right\rangle_{\mathfrak{H}^{\otimes 3}}+3I_{4}(f_{3,T}\tilde{\otimes}_{1}h_{3,T})+12I_{2}(f_{3,T}\tilde{\otimes}_{2}h_{3,T}).$

Simalarly, there exists a constant $C_{2}$ such that

\displaystyle\Psi_{2}(T)

\displaystyle\leq C_{2}\cdot\max_{\begin{subarray}{c}p,q=f,h\\ m,n=1,2,3\\ i=1,\cdots,(m\wedge n)\end{subarray}}\Bigl{(}\left\|p_{m,T}\tilde{\otimes}_{i}q_{n,T}\right\|_{\mathfrak{H}^{\otimes(m+n-2i)}},\left\langle f_{m,T},h_{m,T}\right\rangle_{\mathfrak{H}^{\otimes m}}\Bigr{)}.

(34)

Following from the above result, we can obtain the bound of $\Psi_{3}(T)$ .

We then deal with $\Psi_{1}(T)=\frac{1}{(\mathbb{E}{H_{T}})^{2}}\sqrt{\mathbb{E}[((\mathbb{E}H_{T})^{2}-\left\langle DF_{T},-DL^{-1}F_{T}\right\rangle_{\mathfrak{H}})^{2}]}$ . According to the orthogonality and (32), we have

\displaystyle\Psi_{1}(T)

\displaystyle\leq C_{3}\cdot\max_{\begin{subarray}{c}m,n=1,2,3\\ i=1,\cdots,(m\wedge n)\end{subarray}}\Bigl{(}\left\|f_{m,T}\tilde{\otimes}_{i}f_{n,T}\right\|_{\mathfrak{H}^{\otimes(m+n-2i)}},(E_{T}^{2}-\sum_{i=1}^{m}j!\left\|f_{j,T}\right\|_{\mathfrak{H}^{\otimes j}}^{2})\Bigr{)}.

(35)

Combining formulas (33), (34) and (35), we obtain the desired result. ∎

We now prove the convergence speed of $\hat{k}_{LS}$ . First, we review the CLT of $\hat{k}_{LS}$ .

Theorem 17 ([11] Propositions 4.20).

When $\beta\in(1/2,3/4)$ and 1 holds, $\hat{k}_{LS}$ satisfies the following central limit theorem:

\displaystyle\sqrt{T}(\hat{k}_{LS}-k)

\displaystyle=\frac{\frac{V_{T}}{\sqrt{T}}\hat{\mu}+k\sqrt{T}\hat{\mu}(\hat{\mu}-\mu)-\frac{1}{\sqrt{T}}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}G_{t}}{\frac{1}{T}\int_{0}^{T}V_{t}^{2}\mathop{}\!\mathrm{d}t-(\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}}\stackrel{{\scriptstyle law}}{{\longrightarrow}}\mathcal{N}(0,k\sigma_{\beta}^{2}).

(36)

We then transfrom the above $\sqrt{T}(\hat{k}_{LS}-k)$ as multiple Wiener-Itô integrals.

Proposition 18.

Let $X_{t}=\int_{0}^{t}\mathrm{e}^{-k(t-s)}\mathop{}\!\mathrm{d}G_{s}$ be a OU process driven by $G_{t}$ . $\sqrt{T}(\hat{k}_{LS}-k)$ can be rewritten as:

\sqrt{T}(\hat{k}_{LS}-k)=\frac{I_{0}+I_{1}(f_{1,T})+I_{2}(f_{2,T})}{J_{0}+I_{1}(h_{1,T})+J_{2}(h_{2,T})}.

(37)

where


$\displaystyle I_{0}$	$\displaystyle=\frac{1}{T^{3/2}}\Bigl{(}\mu^{2}a_{T}\cdot d_{T}+q_{T}+\frac{\mu^{2}}{k}a_{T}^{2}+ke_{T}\Bigr{)}+\frac{\mu^{2}}{T^{1/2}}(\mathrm{e}^{-kT}-1),$	(38a)
$\displaystyle f_{1,T}$	$\displaystyle=\frac{\mu}{T^{3/2}}\bigl{(}d_{T}k_{T}-a_{T}l_{T}\bigr{)}+\frac{\mu}{\sqrt{T}}\bigl{(}m_{T}-k_{T}\bigr{)},$	(38b)
$\displaystyle f_{2,T}$	$\displaystyle=\frac{1}{T^{3/2}}l_{T}\otimes k_{T}+\frac{k}{T^{3/2}}l_{T}\otimes l_{T}-\frac{1}{2\sqrt{T}}f_{T},$	(38c)
$\displaystyle J_{0}$	$\displaystyle=\frac{c_{T}}{T}+b_{T}-\frac{1}{T^{2}}(\mu^{2}d_{T}^{2}+e_{T}),$	(38d)
$\displaystyle h_{1,T}$	$\displaystyle=\frac{2\mu}{T}\bigl{(}l_{T}+n_{T}\bigr{)}-\frac{2}{T^{2}}d_{T}l_{T},$	(38e)
$\displaystyle h_{2,T}$	$\displaystyle=g_{T}-\frac{1}{T^{2}}l_{T}\otimes l_{T}.$	(38f)

Proof.

We first deal with the numerator of (36):

\frac{V_{T}}{\sqrt{T}}\hat{\mu}+k\sqrt{T}\hat{\mu}(\hat{\mu}-\mu)-\frac{1}{\sqrt{T}}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}G_{t}.

(39)

According to the definition of above functions, $V_{T}=\mu a_{T}+I_{1}(k_{T})$ . Combining with Proposition 9, we can obtain

	$\displaystyle\frac{V_{T}}{\sqrt{T}}\hat{\mu}$	$\displaystyle=\frac{\mu a_{T}+I_{1}(k_{T})}{T^{1/2}}\frac{\mu d_{T}+I_{1}(l_{T})}{T}$
		$\displaystyle=\frac{1}{T^{3/2}}\bigl{(}\mu^{2}a_{T}\cdot d_{T}+\mu a_{T}I_{1}(l_{T})+\mu d_{T}I_{1}(k_{T})+I_{1}(l_{T})I_{1}(k_{T})\bigr{)}$
		$\displaystyle=\frac{1}{T^{3/2}}\Bigl{(}\mu^{2}a_{T}\cdot d_{T}+\mu a_{T}I_{1}(l_{T})+\mu d_{T}I_{1}(k_{T})$
		$\displaystyle\ \ \ +I_{2}(l_{T}\otimes k_{T})+\left\\|l_{T}\otimes_{1}k_{T}\right\\|_{\mathfrak{H}}^{2}\Bigr{)}.$

Then the second term of (39). Let $Item_{2}:=k\sqrt{T}\hat{\mu}(\hat{\mu}-\mu)$ , we have

	$\displaystyle Item_{2}$	$\displaystyle=k\sqrt{T}(\hat{\mu}-\mu+\mu)(\hat{\mu}-\mu)$
		$\displaystyle=k\sqrt{T}(\hat{\mu}-\mu)^{2}+k\sqrt{T}\mu(\hat{\mu}-\mu)$
		$\displaystyle=\frac{k}{T^{3/2}}\Bigl{(}\frac{\mu^{2}}{k^{2}}(\mathrm{e}^{-kT}-1)^{2}+\frac{2\mu}{k}(\mathrm{e}^{-kT}-1)I_{1}(l_{T})+I_{2}(l_{T}\otimes l_{T})+e_{T}\Bigr{)}$
		$\displaystyle\ \ \ +\frac{k\mu}{T^{1/2}}\Bigl{(}\frac{\mu}{k}(\mathrm{e}^{-kT}-1)+I_{1}(l_{T})\Bigr{)}$
		$\displaystyle=\frac{k}{T^{3/2}}\Bigl{(}\frac{\mu^{2}}{k^{2}}(\mathrm{e}^{-kT}-1)^{2}+\frac{2\mu}{k}(\mathrm{e}^{-kT}-1)I_{1}(l_{T})+I_{2}(l_{T}\otimes l_{T})+e_{T}\Bigr{)}$
		$\displaystyle\ \ \ +\frac{\mu^{2}}{T^{1/2}}(\mathrm{e}^{-kT}-1)+\frac{\mu}{T^{1/2}}\bigl{(}G_{T}-I_{1}(k_{T})\bigr{)}.$

Besides, we can obtain

	$\displaystyle-\frac{1}{\sqrt{T}}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}G_{t}$	$\displaystyle=-\frac{1}{\sqrt{T}}\Bigl{(}\int_{0}^{T}\mu(1-\mathrm{e}^{-kt})\mathop{}\!\mathrm{d}G_{t}+\frac{1}{2}I_{2}(f_{T})\Bigr{)}$
		$\displaystyle=-\frac{1}{\sqrt{T}}\Bigl{(}\mu(G_{T}-I_{1}(m_{T}))+\frac{1}{2}I_{2}(f_{T})\Bigr{)}.$

Next, we consider the denominator. For $\frac{1}{T}\int_{0}^{T}V_{t}^{2}\mathop{}\!\mathrm{d}t$ , we have

	$\displaystyle\frac{1}{T}\int_{0}^{T}V_{t}^{2}\mathop{}\!\mathrm{d}t$	$\displaystyle=\frac{1}{T}\int_{0}^{T}(\mu(1-\mathrm{e}^{-kt})+X_{t})^{2}\mathop{}\!\mathrm{d}t$
		$\displaystyle=\frac{1}{T}\Bigl{(}\int_{0}^{T}\mu^{2}(1-\mathrm{e}^{-kt})^{2}\mathop{}\!\mathrm{d}t+2\int_{0}^{t}\mu(1-\mathrm{e}^{-kt})X_{t}\mathop{}\!\mathrm{d}t+\int_{0}^{T}X_{t}^{2}\mathop{}\!\mathrm{d}t\Bigr{)}$
		$\displaystyle=\frac{1}{T}\Bigl{(}c_{T}+2\mu\int_{0}^{T}\int_{0}^{t}(1-\mathrm{e}^{-kt})\mathrm{e}^{-k(t-s)}\mathop{}\!\mathrm{d}G_{s}\mathop{}\!\mathrm{d}t\Bigr{)}+I_{2}(g_{T})+b_{T}$
		$\displaystyle=\frac{c_{T}}{T}+\frac{2\mu}{T}\bigl{(}I_{1}(l_{T})+I_{1}(n_{T})\bigr{)}+I_{2}(g_{T})+b_{T}.$

Since $\hat{\mu}^{2}=(\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}$ , we can obtain

	$\displaystyle\Bigl{(}\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t\Bigr{)}^{2}$	$\displaystyle=\frac{1}{T^{2}}\Bigl{(}\mu d_{T}+I_{1}(l_{T})\Bigr{)}^{2}$
		$\displaystyle=\frac{1}{T^{2}}\Bigl{(}\mu^{2}d_{T}^{2}+2\mu d_{T}I_{1}(l_{T})+I_{1}^{2}(l_{T})\Bigr{)}$
		$\displaystyle=\frac{1}{T^{2}}\Bigl{(}\mu^{2}d_{T}^{2}+2\mu d_{T}I_{1}(l_{T})+I_{2}(l_{T}\otimes l_{T})+e_{T}\Bigr{)}.$

Combining the above formulas, we obtain the desired result. ∎

For simplicity, let $F_{T}:=(I_{1}(f_{1,T})+I_{2}(f_{2,T}))/(\sqrt{k}\sigma_{\beta}),\;H_{T}:=J_{0}+I_{1}(h_{1,T})+I_{2}(h_{2,T})$ . We show the convergence speed of zero-mean part.

Lemma 19.

Let $Z\sim\mathcal{N}(0,1)$ be a standard Normal variable. Assume $\beta\in(1/2,3/4)$ and 1 holds. When $T$ is large enough, there exists constant $C_{\beta,V}^{\prime}$ such that

\sup_{z\in\mathbb{R}}\left|\mathbb{P}\left(\frac{F_{T}}{H_{T}}\leq z\right)-\mathbb{P}(Z\leq z)\right|\leq\frac{C_{\beta,V}^{\prime}}{T^{\gamma}},

(40)

where $\gamma=\min{\{1/2,3-4\beta\}}$ .

Proof.

According to [11] formulas (9) and (47),

\displaystyle\mathbb{E}H_{T}=\mathbb{E}J_{0}\to\alpha=C_{\beta}\Gamma(2\beta-1)k^{-2\beta}\;\;\;\mathrm{a.s.}.

Combining with [12] Lemma 5.2.4, we can obtain

\displaystyle\max\{\left\|l_{T}\right\|_{\mathfrak{H}}^{2},\left\|n_{T}\right\|_{\mathfrak{H}}^{2}\}\leq C^{(1)}T^{2\beta},

(41)

where $C^{(1)}$ is a constant, and $\left\|h_{1,T}\right\|_{\mathfrak{H}}^{2}\leq C^{(2)}/T^{2-2\beta}$ .

Following from [7] Theorem 1.4 and (41), we have

\displaystyle\left\|g_{T}\right\|_{\mathfrak{H}^{\otimes 2}}\leq\frac{C^{(3)}}{\sqrt{T}},\;\;\;\left\|g_{T}\otimes_{1}g_{T}\right\|_{\mathfrak{H}^{\otimes 2}}\leq\frac{C^{(3)}}{T},\;\;\;\left\|\frac{2}{T^{2}}l_{T}\otimes l_{T}\right\|_{\mathfrak{H}^{\otimes 2}}^{2}\leq\frac{C^{(3)}}{T^{4-4\beta}},

where $C^{(3)}$ is a constant independent of $T$ . Minkowski inequality ensures that $\left\|h_{2,T}\right\|_{\mathfrak{H}^{\otimes 2}}^{2}\leq C^{(4)}/T$ . Simalarly, the Proposition 3.10 of [11] induces that

\displaystyle J_{0}^{2}-\sum_{j=1}^{m}\frac{j!\left\|f_{j,T}\right\|_{\mathfrak{H}^{\otimes j}}^{2}}{k\sigma_{\beta}^{2}}\leq C^{(5)}/T^{3-4\beta},

(42)

and $\left\|f_{1,T}\right\|_{\mathfrak{H}}^{2}\leq C^{(5)}/T$ . The formula (5.13) of[7] ensures that

\displaystyle\left\|f_{2,T}\tilde{\otimes}_{1}h_{2,T}\right\|_{\mathfrak{H}^{\otimes 2}}\leq C^{(5)}/T^{1/2}.

Combining the above results, Theorem 16 and Cauchy-Schwarz inequality, we obtain the Lemma. ∎

The following Lemma shows the bound of non-zero mean part.

Lemma 20.

Assume $\beta\in(1/2,3/4)$ and 1 holds. When $T$ is large enough, there exists constant $C_{1}$ such that

\mathbb{P}\Bigl{(}\left|L_{1}\right|>\frac{C_{1}}{T^{(3/4-\beta)}}\Bigr{)}=0,

where $L_{1}=I_{0}/(\sqrt{k}\sigma_{\beta}H_{T})$ .

Proof.

The Proposition 3.14 and Corollary 3.15 of [11] ensure that when $T$ is large enough,

\frac{1}{T}\int_{0}^{T}V_{t}^{2}\mathop{}\!\mathrm{d}t-\Bigl{(}\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t\Bigr{)}^{2}\to C_{\beta}\Gamma(2\beta-1)k^{-2\beta}=\alpha\;\;\;\mathrm{a.s.}.

(43)

Then we can obtain

\left|L_{1}\right|=\left|\frac{\frac{1}{\sqrt{k}\sigma_{\beta}}I_{0}}{H_{T}}\right|\leq C^{\prime}\left|I_{0}\right|,

where $C^{\prime}$ is a constant. Furthermore, According to (38a), there exists $C^{\prime\prime}$ such that

\left|I_{0}\right|\leq\frac{1}{T^{3/2}}\Bigl{(}\mu^{2}a_{T}\cdot d_{T}+q_{T}+\frac{\mu^{2}}{k}a_{T}^{2}+ke_{T}\Bigr{)}+\frac{\mu^{2}}{T^{1/2}}a_{T}\leq\frac{C^{\prime\prime}}{T^{3/2-2\beta}}.

Combining the above two formulas, we have

\left|L_{1}\right|\leq\frac{C^{(6)}}{T^{3/2-2\beta}}\;\;\;\mathrm{a.s.},

where $C^{(6)}=2C^{\prime}\cdot C^{\prime\prime}$ . Then we obtain the desired result. ∎

We now prove the formula (9).

Proof of formula (9).

According to Lemma 10,

	$\displaystyle\sup_{z\in\mathbb{R}}\left\|\mathbb{P}\Bigl{(}\sqrt{\frac{T}{k\sigma_{\beta}^{2}}}(\hat{k}_{LS}-k)\leq z\Bigr{)}-\mathbb{P}(Z\leq z)\right\|$	$\displaystyle\leq\sup_{z\in\mathbb{R}}\left\|\mathbb{P}\Bigl{(}\frac{F_{T}}{H_{T}}\leq z\Bigr{)}-\mathbb{P}(Z\leq z)\right\|$
		$\displaystyle\ \ \ +\mathbb{P}\Bigl{(}\left\|L_{1}\right\|>\frac{C_{1}}{T^{3/4-\beta}}\Bigr{)}+\frac{1}{\sqrt{2\pi}}\frac{C_{1}}{T^{3/4-\beta}}.$

Combining Lemmas 19 and 20, we obtain the desired result. ∎

Pei et al.[11] show the CLT of least squares estimator $\hat{\mu}_{LS}$ of mean coefficient $\mu$ .

Theorem 21 ([11] Propositions 4.21).

Assume $\beta\in(1/2,1)$ and $G_{t}$ is a self-similar Gaussian process satisfing 1 and $\mathbb{E}[G_{1}^{2}]=1$ . $T^{1-\beta}(\hat{\mu}_{LS}-\mu)$ is asymptotically normal as $T\to\infty$ :

T^{1-\beta}(\hat{\mu}_{LS}-\mu)=\frac{\frac{V_{T}}{T^{\beta}}\big{[}\frac{1}{T}\int_{0}^{T}V_{t}^{2}\mathop{}\!\mathrm{d}t-\mu\hat{\mu}\big{]}-\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}V_{t}\cdot T^{1-\beta}\big{[}\hat{\mu}-\mu\big{]}}{\frac{V_{T}}{T}\cdot\hat{\mu}-\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}V_{t}}\stackrel{{\scriptstyle law}}{{\longrightarrow}}\mathcal{N}(0,1/k^{2}).

(44)

We also transform $T^{1-\beta}(\hat{\mu}_{LS}-\mu)$ as the following multiple integrals.

Proposition 22.

$T^{1-\beta}(\hat{\mu}_{LS}-\mu)$ can be represented by as :

T^{1-\beta}(\hat{\mu}_{LS}-\mu)=\frac{I_{0}^{*}+I_{1}(f_{1,T}^{*})+I_{2}(f_{2,T}^{*})+I_{3}(f_{3,T}^{*})}{J_{0}^{*}+I_{1}(h_{1,T}^{*})+I_{2}(h_{1,T}^{*})},

where


	$\displaystyle I_{0}^{*}=\frac{1}{T^{1+\beta}}\Bigl{(}2\mu\bigl{(}\left\\|k_{T}\otimes_{1}l_{T}\right\\|_{\mathfrak{H}}^{2}+\left\\|k_{T}\otimes_{1}n_{T}\right\\|_{\mathfrak{H}}^{2}\bigr{)}+2k\mu\left\\|n_{T}\otimes_{1}l_{T}\right\\|_{\mathfrak{H}}^{2}+\mu\left\\|m_{T}\otimes_{1}l_{T}\right\\|_{\mathfrak{H}}^{2}\Bigr{)},$		(45a)
	$\displaystyle f_{1,T}^{*}=\frac{1}{T^{1+\beta}}\Bigl{(}(c_{T}-\mu^{2}d_{T}-\frac{\mu^{2}a_{T}}{k}\mathbbm{1}_{[0,T]})+2\mu^{2}a_{T}l_{T}+f_{T}\otimes_{1}l_{T}\Bigr{)}$
	$\displaystyle\;\;\;\;\;\;-\frac{1}{T^{1+\beta}}\frac{\mu^{2}a_{T}}{k}m_{T}+\frac{1}{T^{\beta}}\Bigl{(}b_{T}\mathbbm{1}_{[0,T]}+2g_{T}\otimes_{1}\mathbbm{1}_{[0,T]}\Bigr{)},$		(45b)
	$\displaystyle f_{2,T}^{*}=\frac{1}{T^{1+\beta}}\Bigl{(}2\mu\bigl{(}k_{T}\otimes l_{T}+k_{T}\otimes n_{T}\bigr{)}+2k\mu(n_{T}\otimes l_{T})+\mu m_{T}\otimes l_{T}-\frac{\mu}{2k}a_{T}f_{T}\Bigr{)},$		(45c)
	$\displaystyle f_{3,T}^{*}=\frac{1}{T^{\beta}}\bigl{(}g_{T}\otimes k_{T}+kg_{T}\otimes l_{T}\bigr{)}+\frac{1}{2T^{1+\beta}}f_{T}\otimes l_{T},$		(45d)
	$\displaystyle J_{0}^{*}=\frac{1}{T^{2}}\Bigl{(}\mu^{2}a_{T}\cdot d_{T}+\left\\|l_{T}\otimes_{1}k_{T}\right\\|_{\mathfrak{H}}^{2}\Bigr{)}+\frac{1}{T}\Bigl{(}kc_{T}-k\mu^{2}d_{T}\bigr{)}+kb_{T},$		(45e)
	$\displaystyle h_{1,T}^{*}=\frac{1}{T^{2}}\Bigl{(}\mu a_{T}l_{T}+\mu d_{T}k_{T}\Bigr{)}+\frac{1}{T}\Bigl{(}\mu k_{T}+2k\mu n_{T}+\mu m_{T}\Bigr{)},$		(45f)
	$\displaystyle h_{2,T}^{*}=\frac{1}{T^{2}}l_{T}\otimes k_{T}-\frac{1}{2T}f_{T}+kg_{T}.$		(45g)

Proof.

We first consider the denominator.

\displaystyle\frac{V_{T}}{T}\cdot\hat{\mu}-\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}V_{t}.

According to Proposition 18,

	$\displaystyle\frac{V_{T}}{T}\hat{\mu}$	$\displaystyle=\frac{\mu a_{T}+I_{1}(k_{T})}{T}\frac{\mu d_{T}+I_{1}(l_{T})}{T}$
		$\displaystyle=\frac{1}{T^{2}}\bigl{(}\mu^{2}a_{T}\cdot d_{T}+\mu a_{T}I_{1}(l_{T})+\mu d_{T}I_{1}(k_{T})+I_{1}(l_{T})I_{1}(k_{T})\bigr{)}$
		$\displaystyle=\frac{1}{T^{2}}\Bigl{(}\mu^{2}a_{T}\cdot d_{T}+\mu a_{T}I_{1}(l_{T})+\mu d_{T}I_{1}(k_{T})+I_{2}(l_{T}\otimes k_{T})+\left\\|l_{T}\otimes_{1}k_{T}\right\\|_{\mathfrak{H}}^{2}\Bigr{)}.$

Then, we deal with $-\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}V_{t}$ :

	$\displaystyle-\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}V_{t}$	$\displaystyle=-\frac{1}{T}\Bigl{(}\int_{0}^{T}k\mu V_{t}\mathop{}\!\mathrm{d}t-\int_{0}^{T}kV_{t}^{2}\mathop{}\!\mathrm{d}t+\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}G_{t}\Bigr{)}$
		$\displaystyle=-\frac{k\mu}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t+\frac{k}{T}\int_{0}^{T}V_{t}^{2}\mathop{}\!\mathrm{d}t-\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}G_{t}$
		$\displaystyle=-\frac{k\mu^{2}d_{T}+k\mu I_{1}(l_{T})}{T}+\Bigl{(}\frac{kc_{T}}{T}+\frac{2k\mu}{T}\bigl{(}I_{1}(l_{T})+I_{1}(n_{T})\bigr{)}\Bigr{)}$
		$\displaystyle\ \ \ -\frac{1}{T}\Bigl{(}\mu\bigl{(}G_{T}-I_{1}(m_{T})\bigr{)}+\frac{1}{2}I_{2}(f_{T})\Bigr{)}+kI_{2}(g_{T})+kb_{T}$
		$\displaystyle=\frac{1}{T}\Bigl{(}kc_{T}-k\mu^{2}d_{T}+2k\mu I_{1}(n_{T})+\mu I_{1}(k_{T})+\mu I_{1}(m_{T})+\frac{1}{2}I_{2}(f_{T})\Bigr{)}$
		$\displaystyle\ \ \ +kI_{2}(g_{T})+kb_{T}.$

We next consider the numerator:

\displaystyle\frac{V_{T}}{T^{\beta}}\Big{[}\frac{1}{T}\int_{0}^{T}V_{t}^{2}\mathop{}\!\mathrm{d}t-\mu\hat{\mu}\Big{]}-\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}V_{t}\cdot T^{1-\beta}\big{[}\hat{\mu}-\mu\big{]}.

Since the Proposition 18, we have

	$\displaystyle\frac{V_{T}}{T^{\beta}}\frac{1}{T}\int_{0}^{T}V_{t}^{2}dt$	$\displaystyle=\frac{1}{T^{\beta}}\bigl{(}\mu a_{T}+I_{1}(k_{T})\bigr{)}\Bigl{(}\frac{c_{T}}{T}+\frac{2\mu}{T}\bigl{(}I_{1}(l_{T})+I_{1}(n_{T})\bigr{)}+I_{2}(g_{T})+b_{T}\Bigr{)}$
		$\displaystyle=\frac{1}{T^{1+\beta}}\Bigl{(}\mu a_{T}\cdot c_{T}+2\mu^{2}a_{T}\bigl{(}I_{1}(l_{T})+I_{1}(n_{T})\bigr{)}+c_{T}I_{1}(k_{T})$
		$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+2\mu\bigl{(}I_{2}(k_{T}\otimes l_{T})+I_{2}(k_{T}\otimes n_{T})\bigr{)}$
		$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+2\mu\bigl{(}\left\\|k_{T}\otimes_{1}l_{T}\right\\|_{\mathfrak{H}}^{2}+\left\\|k_{T}\otimes_{1}n_{T}\right\\|_{\mathfrak{H}}^{2}\bigr{)}\Bigr{)}$
		$\displaystyle\;\;\;+\frac{1}{T^{\beta}}\Bigl{(}\mu a_{T}I_{2}(g_{T})+\mu a_{T}b_{T}+I_{3}(k_{T}\otimes g_{T})$
		$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+2I_{1}(k_{T}\otimes_{1}g_{T})+b_{T}I_{1}(k_{T})\Bigr{)}.$

Similarly, we can obtain

	$\displaystyle-\frac{V_{T}}{T^{\beta}}\cdot\mu\hat{\mu}$	$\displaystyle=-\mu\frac{\mu a_{T}+I_{1}(k_{T})}{T^{\beta}}\frac{\mu d_{T}+I_{1}(l_{T})}{T}$
		$\displaystyle=\frac{-\mu}{T^{1+\beta}}(\mu^{2}a_{T}\cdot d_{T}+\mu a_{T}I_{1}(l_{T})+\mu d_{T}I_{1}(k_{T})+I_{1}(l_{T})I_{1}(k_{T}))$
		$\displaystyle=\frac{-\mu}{T^{1+\beta}}\Bigl{(}\mu^{2}a_{T}\cdot d_{T}+\mu a_{T}I_{1}(l_{T})+\mu d_{T}I_{1}(k_{T})$
		$\displaystyle\ \ \ +I_{2}(l_{T}\otimes k_{T})+\left\\|l_{T}\otimes_{1}k_{T}\right\\|_{\mathfrak{H}}^{2}\Bigr{)}.$

Let $Item_{3}=-\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}V_{t}\cdot T^{1-\beta}(\hat{\mu}-\mu)$ , we have

	$\displaystyle Item_{3}$	$\displaystyle=\Bigl{(}\frac{I_{1}(l_{T})}{T^{\beta}}-\frac{\mu a_{T}}{kT^{\beta}}\Bigr{)}\frac{1}{T}\int_{0}^{T}-V_{t}\mathop{}\!\mathrm{d}V_{t}$
		$\displaystyle=\frac{1}{T^{1+\beta}}\biggl{(}kc_{T}I_{1}(l_{T})-k\mu^{2}d_{T}I_{1}(l_{T})+2k\mu I_{2}(n_{T}\otimes l_{T})+2k\mu\left\\|n_{T}\otimes_{1}l_{T}\right\\|_{\mathfrak{H}}^{2}$
		$\displaystyle\ \ \ +\mu I_{2}(k_{T}\otimes l_{T})+\mu\left\\|k_{T}\otimes_{1}l_{T}\right\\|_{\mathfrak{H}}^{2}+\mu I_{2}(m_{T}\otimes l_{T})+\mu\left\\|m_{T}\otimes_{1}l_{T}\right\\|_{\mathfrak{H}}^{2}\biggr{)}$
		$\displaystyle+\frac{k}{T^{\beta}}\Bigl{(}b_{T}I_{1}(l_{T})+I_{3}(g_{T}\otimes l_{T})+2I_{1}(g_{T}\otimes_{1}l_{T})\Bigr{)}$
		$\displaystyle+\frac{1}{2T^{1+\beta}}\Bigl{(}I_{3}(f_{T}\otimes l_{T})+2I_{1}(f_{T}\otimes_{1}l_{T})\Bigr{)}$
		$\displaystyle+\frac{1}{T^{1+\beta}}\Bigl{(}\mu^{3}a_{T}\cdot d_{T}-\mu a_{T}\cdot c_{T}-2\mu^{2}a_{T}I_{1}(n_{T})\Bigr{)}$
		$\displaystyle-\frac{1}{kT^{1+\beta}}\Bigl{(}\mu^{2}a_{T}I_{1}(k_{T})+\mu^{2}a_{T}I_{1}(m_{T})+\frac{1}{2}\mu a_{T}I_{2}(f_{T})\Bigr{)}$
		$\displaystyle-\frac{1}{T^{\beta}}\bigl{(}\mu a_{T}\cdot b_{T}+\mu a_{T}I_{2}(g_{T})\bigr{)}.$

Combining the above formulas, we obtain the Proposition. ∎

Denote $F_{T}^{*}:=I_{0}^{*}+I_{1}(f_{1,T}^{*})+I_{2}(f_{2,T}^{*})+I_{3}(f_{3,T}^{*})$ and $H_{T}^{*}=\frac{1}{k}(J_{0}^{*}+I_{1}(h_{1,T}^{*})+I_{2}(h_{1,T}^{*}))$ . We now prove the upper bounds of zero-mean part.

Lemma 23.

Let $Z\sim\mathcal{N}(0,1)$ be a standard Normal variable. Assume $\beta\in(1/2,1)$ and $G_{t}$ is a self-similar Gaussian process satisfing 1 and $\mathbb{E}[G_{1}^{2}]=1$ . When $T$ is large enough, there exists a constant $C_{\beta,V}^{\prime}$ such that

\sup_{z\in\mathbb{R}}\left|\mathbb{P}\left(\frac{F_{T}^{*}}{H_{T}^{*}}\leq z\right)-\mathbb{P}(Z\leq z)\right|\leq\frac{C_{\beta,V}^{\prime}}{T^{2-2\beta}},

(46)

Proof.

According to Lemma 19, (45f) and (45g), we have

\displaystyle\left\|h_{1,T^{*}}\right\|_{\mathfrak{H}}^{2}\leq\frac{K_{1}}{T^{2-2\beta}},\;\;\;\left\|h_{2,T^{*}}\right\|_{\mathfrak{H}^{\otimes 2}}^{2}\leq\frac{K_{1}}{T},

where $K_{1}$ is a constant independent of $T$ . It is easy to see that Lemma 19 and equation (45e) implies $\mathbb{E}H_{T}^{*}\to\alpha\;\mathrm{a.s.}$ . Furthermore, when $T$ is large enough, there exists constant $K_{2}$ such that

\displaystyle\left|(\mathbb{E}H_{T}^{*})^{2}-\alpha^{2}\right|

\displaystyle\leq\frac{K_{2}}{T^{2-2\beta}}.

(47)

Since $G_{t}$ is self-similar, Lemma 19 implies $\left|\left\|f_{1,T}^{*}\right\|_{\mathfrak{H}}^{2}-\alpha^{2}\right|\leq\frac{K_{3}}{T^{2-2\beta}}$ . Also, we can obtain

\displaystyle\left\|f_{2,T}^{*}\right\|_{\mathfrak{H}^{\otimes 2}}^{2}\leq\frac{K_{3}}{T^{2}},\;\;\;\left\|f_{3,T}^{*}\right\|_{\mathfrak{H}^{\otimes 3}}^{2}\leq\frac{K_{3}}{T}

(48)

Combining above three formulas, we have

\displaystyle J_{0}^{*,2}-\sum_{j=1}^{m}j!\left\|f_{j,T}^{*}\right\|_{\mathfrak{H}^{\otimes j}}^{2}\leq K_{4}/T^{2-2\beta},

Following from Theorem 16 and Cauchy-Schwarz inequality, we obtain the result. ∎

We next consider the non-zero mean part.

Lemma 24.

Assume $\beta\in(1/2,1)$ and $G_{t}$ is a self-similar Gaussian process satisfing 1 and $\mathbb{E}[G_{1}^{2}]=1$ . Let $M_{1}=I_{0}^{*}/H_{T}^{*}$ . When $T$ is large enough, there exists a constant $C_{1}$ independent of $T$ such that

\mathbb{P}\Bigl{(}\left|M_{1}\right|>\frac{C_{1}}{T^{1/2-\beta/2}}\Bigr{)}=0.

Proof.

Following from Lemma 23, we have $\left|M_{1}\right|\leq C^{\prime}\left|I_{0}^{*}\right|$ , where $C^{\prime}$ is a constant independent of $T$ . Furthermore, Equation (45a) implies that there exists $C^{\prime\prime}$ such that

\displaystyle\left|I_{0}^{*}\right|\leq\frac{C^{\prime\prime}}{T^{1-\beta}},\;\;\;\left|M_{1}\right|\leq\frac{C_{1}}{T^{1-\beta}}\;\;\;\mathrm{a.s.},

where $C_{1}=2C^{\prime}\cdot C^{\prime\prime}$ . Combining the above formulas, we obtain the desired result. ∎

We now prove the formula (11).

Proof of formula (11).

Following from Lemma 10, we have

	$\displaystyle\sup_{z\in\mathbb{R}}\left\|\mathbb{P}\Bigl{(}\frac{k}{T^{\beta-1}}(\hat{\mu}_{LS}-\mu)\leq z\Bigr{)}-\mathbb{P}(Z\leq z)\right\|$	$\displaystyle\leq\sup_{z\in\mathbb{R}}\left\|\mathbb{P}\Bigl{(}\frac{F_{T}^{}}{H_{T}^{}}\leq z\Bigr{)}-\mathbb{P}(Z\leq z)\right\|$
		$\displaystyle\ \ \ +\mathbb{P}\Bigl{(}\left\|M_{1}\right\|>\frac{C_{1}}{T^{1/2-\beta/2}}\Bigr{)}$
		$\displaystyle\ \ \ +\frac{1}{\sqrt{2\pi}}\frac{C_{1}}{T^{1/2-\beta/2}}.$

Combining Lemmas 23 and 24, we obtain the formula (11). ∎

Acknowledgments

We gratefully acknowledge the very valuable suggestions by referees. Y. Chen is supported by National Natural Science Foundation of China (NSFC) with grant No.11961033.

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	$\displaystyle\left\\|\phi\right\\|_{\mathfrak{H}_{1}}^{2}$	$\displaystyle=C_{\beta}\int_{[0,T]^{2}}\phi(r_{1})\phi(r_{2})\left\|r_{1}-r_{2}\right\|^{2\beta-2}\mathop{}\!\mathrm{d}r_{1}\mathop{}\!\mathrm{d}r_{2},$
	$\displaystyle\left\\|\phi\right\\|_{\mathfrak{H}_{2}}^{2}$	$\displaystyle=C_{\beta}^{\prime}\int_{[0,T]^{2}}\left\|\phi(r_{1})\phi(r_{2})\right\|(r_{1}r_{2})^{\beta-1}\mathop{}\!\mathrm{d}r_{1}\mathop{}\!\mathrm{d}r_{2}.$

	$\displaystyle\left\|\left\\|\phi\right\\|_{\mathfrak{H}^{\otimes{2}}}^{2}-\left\\|\phi\right\\|_{\mathfrak{H}_{1}^{\otimes{2}}}^{2}\right\|$	$\displaystyle\leq\left\\|\phi\right\\|_{\mathfrak{H}_{2}^{\otimes{2}}}^{2}+2C_{\beta}^{\prime}\left\\|K\varphi\right\\|_{\mathfrak{H}_{1}}^{2},$
	$\displaystyle\left\|\langle\varphi,\psi\rangle_{\mathfrak{H}^{\otimes{2}}}-\langle\varphi,\psi\rangle_{\mathfrak{H}_{1}^{\otimes{2}}}\right\|$	$\displaystyle\leq\left\|\langle\varphi,\psi\rangle_{\mathfrak{H}_{2}^{\otimes{2}}}\right\|+2C_{\beta}^{\prime}\left\|\langle K\varphi,K\psi\rangle_{\mathfrak{H}_{1}}\right\|.$

$\displaystyle\left\|B(z)\right\|$	$\displaystyle=\left\|\mathbb{P}\biggl{(}\sqrt{\frac{kT}{\alpha^{2}\sigma_{\beta}^{2}}}\biggl{[}\frac{1}{T}\bigl{(}\int_{0}^{T}V^{2}_{t}\mathop{}\!\mathrm{d}t-(\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}\bigr{)}-\alpha\biggr{]}\geq\nu\biggr{)}-\mathbb{P}(Z\leq z)\right\|$
	$\displaystyle\leq\left\|\mathbb{P}\biggl{(}\sqrt{\frac{kT}{\alpha^{2}\sigma_{\beta}^{2}}}\biggl{[}\frac{1}{T}\bigl{(}\int_{0}^{T}V^{2}_{t}\mathop{}\!\mathrm{d}t-(\frac{1}{T}\int_{0}^{T}V_{t}\mathop{}\!\mathrm{d}t)^{2}\bigr{)}-\alpha\biggr{]}\geq\nu\biggr{)}-\overline{\Phi}(\nu)\right\|$
	$\displaystyle\ \ \ +\left\|\overline{\Phi}(\nu)-\mathbb{P}(Z\leq z)\right\|$
	$\displaystyle=\left\|D(\nu)\right\|+\left\|\overline{\Phi}(\nu)-\mathbb{P}(Z\leq z)\right\|.$	(26)

	$\displaystyle\mathbb{P}\biggl{(}\left\|\sqrt{\frac{kT}{\alpha^{2}\sigma_{\beta}^{2}}}I_{4}\right\|>T^{-1/3}\biggr{)}$	$\displaystyle\leq\mathbb{P}\biggl{(}\left\|K_{1}\frac{\mu\cdot\mathrm{e}^{-kT}\cdot Z_{T}}{k}\right\|>T^{1/6}\biggr{)}$
		$\displaystyle\ \ \ +\mathbb{P}\biggl{(}\left\|K_{1}\frac{\mu\cdot M_{T}}{k}\right\|>T^{1/6}\biggr{)}$
		$\displaystyle\ \ \ +\mathbb{P}\biggl{(}\left\|K_{1}\frac{\mu\cdot(2-2\mathrm{e}^{-kT})\cdot F_{T}}{k}\right\|>T^{7/6}\biggr{)}$
		$\displaystyle\ \ \ +\mathbb{P}\biggl{(}\left\|K_{1}\cdot F^{2}_{T}\right\|>T^{7/6}\biggr{)}.$

	$\displaystyle\sup_{z\in\mathbb{R}}\left\|\mathbb{P}\Bigl{(}\frac{k}{T^{\beta-1}}(\hat{\mu}_{LS}-\mu)\leq z\Bigr{)}-\mathbb{P}(Z\leq z)\right\|$	$\displaystyle\leq\sup_{z\in\mathbb{R}}\left\|\mathbb{P}\Bigl{(}\frac{F_{T}^{}}{H_{T}^{}}\leq z\Bigr{)}-\mathbb{P}(Z\leq z)\right\|$
		$\displaystyle\ \ \ +\mathbb{P}\Bigl{(}\left\|M_{1}\right\|>\frac{C_{1}}{T^{1/2-\beta/2}}\Bigr{)}$
		$\displaystyle\ \ \ +\frac{1}{\sqrt{2\pi}}\frac{C_{1}}{T^{1/2-\beta/2}}.$