Asymptotically optimal Wasserstein couplings for the small-time stable domain of attraction

In preparation for the summary of our results in Table 1, we introduce some notation: $f(t)\lesssim g(t)$ as $t\downarrow 0$ holds for two functions $f,g\geqslant 0$ if there exists $c,t_{0}>0$ satisfying $f(t)\leqslant cg(t)$ for all $t\in(0,t_{0}]$. An eventually positive function $G$ is slowly varying at infinity, $G\in\mbox{SV}_{\infty}$, if $\lim_{x\to\infty}G(cx)/G(x)=1$ for all $c>0$.

Table 1 summarises our results on the convergence rates established here for processes in both domains of attraction of stable processes. Recall that an $\alpha$-stable process has a finite $q$-moment if and only if $q<\alpha$. Due to this technical constraint, our upper bounds on the $L^{q}$-Wasserstein distance, defined in (1) below, always require $q<\alpha$ for the corresponding domains of attraction. We remark that, in this case, both lower and upper bounds are typically asymptotically equivalent up to a multiplicative constant, making our methods and couplings, rate optimal. Indeed, in the domain of normal attraction, this occurs for any admissible $q>0$ if $\alpha\in(0,1)$ and for the $L^{1}$-Wasserstein distance if $\alpha\in(1,2)$. Our bounds for the domain of non-normal attraction are also seen to be rate optimal when $G$ is sufficiently regular (see discussion following Theorem 2.3 and Corollary 2.4 below) when $q=1$ (and hence $\alpha>1$).

More precisely, in Table 1 we let $\bm{X}=(\bm{X}_{t})_{t\in[0,1]}$ be a Lévy process in $\mathbb{R}^{d}$ attracted to an $\alpha$-stable process $\bm{Z}=(\bm{Z}_{t})_{t\in[0,1]}$ with normalising function $g$. That is, $\bm{X}^{t}=(\bm{X}_{s}^{t})_{s\in[0,1]}\coloneqq(\bm{X}_{st}/g(t))_{s\in[0,1]}$, $t\in(0,1]$, satisfies $\bm{X}^{t}_{1}=\bm{X}_{t}/g(t)\xrightarrow{d}\bm{Z}_{1}$ as $t\downarrow 0$. The table gives asymptotic bounds on the distance $\mathcal{W}_{q}(\bm{X}^{t},\bm{Z})$ as $t\downarrow 0$ in both regimes of attraction. We let the assumptions of either Theorem 2.1 (with $p=1$, for the domain of normal attraction) or Theorem 2.3 (for the domain of non-normal attraction) hold for $\alpha\in(0,2)\setminus\{1\}$ and pick $q\in(0,\alpha)\cap(0,1]$ satisfying $\mathds{E}[|\bm{X}_{1}|^{q}]<\infty$. We stress that the lower bounds in Table 1 in both domains of normal and non-normal attraction require no assumptions beyond the existence of the scaling limit (see Theorem 5.1 below for the precise description of the class of Lévy processes $\bm{X}$ attracted to a stable process $\bm{Z}$). In particular, as explained in the caption of Table 1, for any $\bf{X}$ in the domain of non-normal attraction and arbitrary $\epsilon>0$, there exists a positive increasing sequence $(t_{k})_{k\in\mathbb{N}}$ tending to infinity, such that the lower bound on the $L^{q}$-Wasserstein distance satisfies $L(t_{k})\geqslant|\log t_{k}|^{-1-\epsilon}$ for all $k\in\mathbb{N}$. In Example 3.2 below, we show that, even if the slowly varying function $G(1/t)=g(t)t^{-1/\alpha}$ in the scaling limit grows arbitrarily slowly, the lower bound $L$ may be asymptotically equivalent to it and bounded below by $1/\log^{1+\varepsilon}(1/t)$ for all small $t>0$.

Recall that the slowly varying function $t\mapsto G(1/t)$ in the scaling limit $\bm{X}_{t}/g(t)\xrightarrow{d}\bm{Z}_{1}$ (as $t\downarrow 0$) is uniquely determined up to asymptotic equivalence only. Interestingly, our results imply that the rate of convergence in the Wasserstein distance can be affected by different choices of $G$, see Remark 2.5(IV) below for more details. Finally, we note that the couplings yielding the upper bounds in Table 1 require some structural assumptions on the Lévy measure of $\bf{X}$ discussed in Sections 2 and 5 below.

One of the main purposes of this article is to introduce two couplings between two arbitrary multivariate Lévy processes $\bm{X}$ and $\bm{Y}$ and analyse their properties. Both coupling constructions are centered around coupling the respective Poisson jump measures $\Xi_{\bm{X}}$ and $\Xi_{\bm{Y}}$. As the Brownian components of $\bm{X}$ and $\bm{Y}$ are coupled synchronously under both couplings, the couplings are named after the techniques involved in coupling of the Poisson jump measures $\Xi_{\bm{X}}$ and $\Xi_{\bm{Y}}$: the first is the thinning coupling, as it is based on Poisson thinning (see Subsection 4.1), and the second is the comonotonic coupling, based on the minimal transport coupling of real-valued random variables and LePage’s simulation method (see Subsection 4.2). As illustrated in Figure 1, the thinning coupling aims to maximise the intersection of the Poisson jump measures, whereas the comonotonic coupling aims to establish an optimal one-to-one correspondence between the atoms of the Poisson jump measures.

The assumptions and constructions of both couplings are rather different. In the thinning coupling, we consider a common dominating Lévy measure (say, the sum of both Lévy measures) such that the Lévy measures of both processes are absolutely continuous with respect to it with a bounded density. Then, we consider a Poisson measure with mean measure given by the dominating Lévy measure and then thin the Poisson measure appropriately to produce coupled Poisson measures with mean measures given by the Lévy measures of the processes. This maximises the common jumps of both processes.

For the comonotonic coupling, we assume that both processes have a radial decomposition with their own angular measures. We then construct a radial decomposition for both with respect to a common angular measure. We use this measure and LePage’s method to construct a one-to-one correspondence of jumps in which both processes jump in the same direction but with different magnitudes. Indeed, both Poisson measures are a transformation of a standard Poisson measure with independent decorations that select the direction of the jump. With our assumption, we construct such a transformation explicitly with the following properties. The decorations of both processes agree. Conditionally given a direction, the jumps of both processes, when ordered by decreasing magnitude, are in a one-to-one correspondence that mimics the relationship of real random variables under the minimal transport (or comonotonic) coupling. More precisely, the magnitudes are expressed as right inverse of the radial tail Lévy measure evaluated on the epochs of a standard Poisson process.

Few results identifying the small-time convergence rate exist in the multivariate setting even when the limit is Gaussian. Indeed, most results in this regime are restricted to dimension $1$ and often require finite jump activity [11, 10, 14, 15]. In those situations, the limit law of the rescaled error can be identified for some functionals, leading to accurate estimates of the resulting bias and the celebrated continuity corrections [10, 15].

For heavy-tailed stable limits (i.e. non-Gaussian) and infinite activity Lévy processes attracted to the Gaussian law, again in one dimension, the literature is more scarce and only a fraction of the analogous results exist (see [7] for the convergence of certain path statistics to heavy-tailed limits). There are several complications in developing such results for small-time. First, the Berry–Esseen type bounds (see e.g. [24, 37]), commonly used to establish convergence rates to the Gaussian law, fail to give convergent upper bounds since the jump intensity is vanishing in the small-time regime. Second, the rescaled variables often either fail to be uniformly integrable or their uniform integrability is difficult to prove (see, e.g. [7]).

In [16], the authors consider estimating the density of a discretely observed Lévy process satisfying Orey’s condition. Under the assumption that sufficiently large jumps are identifiable and removable in the sample, the estimation attains a minimax rate that is optimal up to a logarithmic factor if the Blumenthal–Getoor index is known. This regime is different from our situation, as the authors assume that we may remove all sufficiently large jumps. In fact, under this kind of assumption, the residual small-jump process may not be attracted to a stable process but to a Brownian motion [2].

In [34, 30], the authors introduce couplings between Lévy processes to bound the Wasserstein distance between them. The coupling in [34] is generic and pays special attention to the small jumps. However, the bounds fail to converge to zero when applied to a stable process and a Lévy process in its small-time stable domain of attraction. In contrast to the coupling used in [34], where the authors couple the big-jump components based on the magnitude of the jumps (i.e. based on a common threshold), we couple these components matching their jump intensities. Moreover, in [34] the authors couple the small jumps through an artificial Brownian motion, while we instead couple the compensated Poisson measures directly. On the other hand, the coupling in [30], based on McCann’s coupling and Rogers’ results on random walks, is the optimal Markovian coupling. However, the coupling requires such tight control of the infinitesimal dynamics of the processes that the coupling could only be constructed for Lévy processes with finitely many jumps on compact intervals, excluding all heavy-tailed stable processes and most processes in the small-time domain of attraction of Brownian motion.

Although the slow convergence phenomenon under the presence of a slowly varying function that does not converge to a positive constant has been observed in some specific settings such as in the case of Hill’s estimator (see [17, p. 193–195]), to the best of our knowledge it was first documented rigorously in [9] in an elementary general setting. The authors in [9] lower bound the Prokhorov distance between the marginals of the limit and that of a random walk in its domain of non-normal attraction with a function $b(n)$, satisfying $\limsup_{n\to\infty}b(n)\log^{1+\varepsilon}(n)=\infty$ for any $\varepsilon>0$. However, as is often the case with lower bounds in the form of upper limits, the sparsity of the sequence of times along which the divergence holds remains unclear. The present paper extends the applicability of such a lower bound and strengthens the conclusions. In particular we show that the function analogous to $b(n)$ is typically slowly varying and provide some explicit asymptotically equivalent lower bounds.

In Section 2 we introduce the main results of the paper, namely, upper and lower bounds on the convergence rate for processes in the domains of normal and non-normal attraction. Subsection 2.5 explains why the existing literature cannot be directly applied to obtain the bounds presented in Section 2. We present two examples in Section 3 in which our main results are applied to tempered stable processes. The two couplings for general Lévy processes on $\mathbb{R}^{d}$ used to prove the upper bounds on the Wasserstein distance in Section 2 are introduced in Section 4. General upper bounds (in $L^{p}$) for each component in the Lévy–Itô decomposition of coupled Lévy processes are also established in Section 4. The upper bounds for processes in the domain of (normal and non-normal) attraction of a stable process (Gaussian and heavy-tailed) are established in Section 5, while the lower bounds are established in Section 6. The proofs of the results stated in Section 2 are given in Section 7. Section 8 concludes the paper.

For a Lévy process $\bm{X}$ to be in the domain of attraction of an $\alpha$-stable process $\bm{Z}$, the necessary condition in (25) of Theorem 5.1 suggests the Lévy measure of $\bm{X}$ around the origin should be “asymptotically absolutely continuous” with respect to the $\alpha$-stable Lévy measure of $\bm{Z}$ (see also Remark 5.3(b) below). Assumption ( (T).) quantifies the regularity of the corresponding density at the origin $\bm{0}$ via the parameter $p>0$ (the larger $p$ is, the more asymptotic regularity there is). Assumption ( (T).) is required for the upper bound on the rate of convergence in the scaling limit in Theorem 2.1 and is stated in Section 5.2 below. Moreover, Assumption ( (T).), widely satisfied in practice (e.g. in the class of tempered stable processes [39] with $p=1$; cf. Section 3 below for specific examples), can be seen as quantifying the speed of convergence in the necessary condition (25) for $\bm{X}$ to be in the stable domain of attraction.

Throughout the paper, for positive functions $f_{1}$ and $f_{2}$, we use the notation $f_{1}(x)=\mathcal{O}(f_{2}(x))$ as $x\downarrow 0$ if $\limsup_{x\downarrow 0}f_{1}(x)/f_{2}(x)<\infty$, and $f_{1}(x)=\mathrm{o}(f_{2}(x))$ as $x\downarrow 0$ if $\lim_{x\downarrow 0}f_{1}(x)/f_{2}(x)=0$.

Consider the case where the slowly varying function $G$ in the scaling limit $(\bm{X}_{st}/(t^{1/\alpha}G(1/t)))_{s\in[0,1]}\xrightarrow{d}(\bm{Z}_{s})_{s\in[0,1]}$, as $t\downarrow 0$, is not asymptotically equivalent to a positive constant. In this section, we show that the lower bound on the $\mathcal{W}_{q}$-distance cannot be upper bounded by a positive non-decreasing function $\phi$ satisfying $\int_{0}^{1}\phi(t)t^{-1}\mathrm{d}t<\infty$. The lower bound requires no assumptions (beyond $\bm{X}$ being in domain of attraction), while the assumptions for the upper bounds give us multiplicative non-asymptotic control over the distance from $G(x/t)/G(1/t)$ to $1$ for small $t>0$ and any $x>0$.

The upper bound in Theorem 2.3(a) below require an additional technical Assumption ( (C).), see Section 5 below. Intuitively, these assumptions require non-parametric structural properties of the Lévy measure of $\bm{X}$ that allows us to compare it to the Lévy measure of the stable limit $\bm{Z}$. Indeed, the necessary condition in (25) of Theorem 5.1 suggests the Lévy measure of $\bm{X}$ around the origin should “asymptotically admit a radial decomposition that is close to that of the stable process”. Assumption ( (C).) states precisely this and specifies the proximity of the corresponding radial decomposition to that of the stable process via the parameters $p,\delta>0$ (as before, the larger $p$ and $\delta$ are, the closer the radial decompositions are). Moreover, both conditions are widely satisfied with $p=1$, e.g. for the class of tempered $\alpha$-stable processes (see [39] and, for specific examples, Section 3 below).

Since $G_{2}$ bounds $\left|G(1/(2t))/G(1/t)-1\right|\leqslant G_{1}(1/2)G_{2}(t)$ for all small $t>0$ and $G$ is not asymptotically equivalent to a positive finite constant, Lemma 7.2 below (which extends [9, Prop., p. 683]) implies $G_{2}$ cannot be upper bounded by any non-decreasing function $\phi$ satisfying $\int_{0}^{1}\phi(t)t^{-1}\mathrm{d}t<\infty$. The assumption on the slow variation of $G_{2}$ in Theorem 2.3(a) is not essential and may be replaced by assuming that $G_{2}$ dominates any (positive) power at zero. However, by Lemma 7.1 below, in most cases of interest such a function $G_{2}$ will be slowly varying.

Given a slowly varying function $G$, the construction of functions $G_{1}$ and $G_{2}$ satisfying Assumption ( (S).) is not immediately clear. However, in most cases and for a sufficiently regular $G$, by virtue of Lemma 7.1, Assumption ( (S).) will be satisfied by choosing $G_{2}(t)\sim t|G^{\prime}(1/t)|/G(1/t)$ as $t\downarrow 0$ and a slowly varying $G_{1}$ (at $0$ and $\infty$) with $G_{1}(x)\geqslant|\log x|\cdot\sup_{t>0,y\in[x\wedge 1,x\vee 1]}G^{\prime}(y/t)/G^{\prime}(1/t)$ (for $a,b\in\mathbb{R}$, we denote $a\wedge b\coloneqq\min\{a,b\}$). In such cases, the lower bound in Theorem 2.3 is (by Lemma 7.1) proportional to $G_{2}$, i.e. $G_{2}(t)\sim|1-a(t)|/\log 2$ as $t\downarrow 0$, making the comonotonic coupling rate optimal with respect to the $\mathcal{W}_{1}$-distance when $\alpha>1$. The following corollary makes this precise and shows that this is the case for a large class of processes in the domain of non-normal attraction.

The main idea behind the proof of Theorems 2.1 & 2.3 is a good coupling between $\bm{X}$ and $\bm{Z}$. The two couplings we apply in this article, are the thinning coupling and the comonotonic coupling introduced in Sections 4.1 & 4.2. In Theorem 2.3 we solely apply the comonotonic coupling, since this yields clear and concise results. Note that one could apply the thinning coupling to get a similar result in the domain of non-normal attraction. However, since this would require a lengthy argument, and would distort the main story and result, this has been left out of the paper. In comparison, it is easier to use the comonotonic coupling to give bounds for processes in the domain of normal attraction.

The domain of attraction to Brownian motion is substantially different as the previously described couplings are inapplicable. Obtaining a coupling between Brownian motion and other Lévy processes that reduced the $L^{p}$-distance in uniform norm has been the work of a large body of literature (which we review in Subsection 2.5 below). In this paper, we use a simple independent coupling, which, heuristically, compares the pure-jump component of $\bm{X}$ with the null process $\bm{0}$. Let $\langle\cdot,\cdot\rangle$ denote the Euclidean inner product on $\mathbb{R}^{d\times d}$, $\bm{0}$ denote the zero-vector in $\mathbb{R}^{d}$ as well as the zero-matrix in $\mathbb{R}^{d\times d}$ and let $\mathbb{R}^{d}_{\bm{0}}\coloneqq\mathbb{R}^{d}\setminus\{\bm{0}\}$. Let $\varphi_{\bm{X}}(\bm{u})\coloneqq\mathds{E}[e^{i\langle\bm{u},\bm{X}\rangle}]$ for $\bm{u}\in\mathbb{R}^{d}$ denote the characteristic function of $\bm{X}$. Furthermore, let $\psi_{\bm{S}}$ be the Lévy-Khintchine exponent of $\bm{S}$, given by $\psi_{\bm{S}}(\bm{u})\coloneqq t^{-1}\log\varphi_{\bm{S}_{t}}(\bm{u})$ for $\bm{u}\in\mathbb{R}^{d}$ and $t>0$.

Parts (a) and (c) of Theorem 2.8, with $q=1$, imply that for processes whose pure jump part is in the domain of attraction of a $\beta$-stable process, the upper and lower bounds are essentially proportional to $t^{1/\max\{1,\beta\}-1/2}$ and $t^{1-\max\{1,\beta\}/2}$, respectively. These agree in the finite variation case with rate $\sqrt{t}$ and also as $\beta\to 2$ with an arbitrarily deteriorating convergence rate. As shown in Figure 2 these bounds are not far from each other for fixed $\beta$, and the powers of $t$ from the rates are also not far. In the ‘limiting case’ where $\bm{S}$ is itself attracted to a Brownian motion and $\beta=2$, the rescaled process $(\bm{S}_{st}/\sqrt{t})_{s\in[0,1]}$ is distributionally close to $\ell(t)\bm{B}$ (see e.g. [26, Thm 2]) for a slowly varying function $\ell$ satisfying $\lim_{t\downarrow 0}\ell(t)=0$. It is thus natural to expect that the convergence, in this case, is slow as in Theorem 2.3 above, see Example 6.7.

The couplings and methods used to achieve these bounds are crucial, and they differ significantly from the classical methods used to find rates of convergence. Indeed, if we tried to use standard methods (namely, the Berry–Esseen theorem or [34]) to construct bounds for small-time domain of attraction, the bounds would not converge as $t\downarrow 0$.

The Berry–Essen theorem exploits an increase in the activity of the process to obtain bounds on the distance between a random walk and the limit law. In the small-time regime, the activity is instead decreasing, explaining the unsuitability of this tool in this context (see details below). In fact, the bound would converge to infinity as $t\downarrow 0$ (and in particular does not go to $0$) unlike the bounds introduced in this paper. The coupling in [34] couples corresponding components of the Lévy–Itô decompositions for a common small-jump cutoff level. When the time horizon is fixed and the Lévy measure is supported on $[-\varepsilon,\varepsilon]$, the bounds of [34] are asymptotically sharp as $\varepsilon\downarrow 0$. However, for general Lévy measures and as time tends to $0$, no time-dependent cutoff level $\varepsilon_{t}$ can be used to obtain convergent bounds. The lack of convergent bounds in the small-time domain of attraction of stable processes is mainly caused by a difference in the jump intensities of the large-jump components (see details below).

We first explain why the Berry–Esseen theorem does not yield suitable bounds even when the limit is Gaussian (see [18, 22]). For the explanation, it is enough to consider the one-dimensional case. Let $(X_{t})_{t\in[0,1]}$ be a zero-mean Lévy process on $\mathbb{R}$ with characteristic triplet $(\gamma,\sigma^{2},\nu_{X})$ (see [40, Def. 8.2]) and finite fourth moment. The variance of $X$ is given by $\mathds{E}[X_{t}^{2}]=(\sigma^{2}+\mu_{2})t$, where $\mu_{2}\coloneqq\int_{\mathbb{R}\setminus\{0\}}x^{2}\nu_{X}(\mathrm{d}x)$. Then, $X_{1}^{t}=X_{t}/\sqrt{(\sigma^{2}+\mu_{2})t}$ is attracted to a standard Gaussian random variable $Z$ as $t\downarrow 0$. Denote by $\nu_{t}$ the Lévy measure of $X_{1}^{t}$, the Berry–Esseen theorem thus implies that there exists some universal constant $C>0$, such that

for all $t\in(0,1]$. As we can see above, this upper bound will tend to $\infty$ as $t\downarrow 0$ and is therefore not an informative bound in the small-time regime.

For Lévy processes in the domain of attraction of an $\alpha$-stable law, an application of the bounds in [34] does not yield convergent bounds. The proofs of the bounds in [34] rely on the coupling of small jumps to a Gaussian law. Again, it is enough to consider the one-dimensional case. Let $X$ be symmetric and in the domain of attraction of the symmetric $\alpha$-stable random variable $Z$ with $\alpha>1$. Suppose their Lévy measures satisfy $\nu_{X}(\mathbb{R}\setminus(-x,x))=x^{-\alpha}+x^{-(\alpha+1)/2}$ and $\nu_{Z}(\mathbb{R}\setminus(-x,x))=x^{-\alpha}$ for $x>0$. In this case we have $X^{t}_{1}=X_{t}/t^{1/\alpha}\xrightarrow{d}Z$ as $t\downarrow 0$.

Let $\eta=(\alpha-1)/2>0$ and apply [34, Thm 11] (at time $1$ and cutoff $\varepsilon_{t}$) to obtain:

where we used the formula for the $L^{1}$-Wasserstein distance in [19, p. 8]. For the first line in the display above to vanish as $t\downarrow 0$, we require $\varepsilon_{t}=\mathrm{o}(1)$. The term in the middle line of the display above equals