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We consider a $d$-dimensional stochastic differential equation (SDE) of the form $dU_t = b(U_t)d t + \sigma d Z_t$ with initial condition $U_0 = u\in \mathbb{R}^d$ and additive noise. Denote by $X_t^x$ the (unique) solution with initial value $X_0^x = x$, if the driving noise $Z_t$ is a $d$-dimensional rotationally symmetric $\alpha$-stable ($1<\alpha<2$) L\'evy process, and by $Y_t^y$ the (unique) solution with initial value $Y_0^y=y$, if the driving noise is a $d$-dimensional Brownian motion. We give precise estimates of the total variation distance $\|{\rm law} (X_{t}^x)-{\rm law} (Y_{t}^y)\|_{TV}$ for all $t>0$, and we show that the ergodic measures $\mu_\alpha$ and $\mu_2$ of $X_t^x$ and $Y_t^y$, respectively, satisfy $\|\mu_\alpha-\mu_2\|_{TV} \leq 2\wedge\left[\frac{Cd^{3/2}}{\alpha-1}(2-\alpha)\right]$. As an application, we consider the following $d$-dimensional Poisson PDE \begin{gather*} -(-\Delta)^{\alpha/2} f(x) + \langle b(x), \nabla f(x)\rangle = h(x)-\mu_\alpha(h), \end{gather*} where $h \in C_b(\mathbb{R}^d,\mathbb{R})$ and $1<\alpha\leq 2$. We show that this PDE admits a solution $f_\alpha$, which is continuous, but may have linear growth. For fixed $\alpha_0\in(1,2)$, we show that for any $\alpha_0\leq \alpha <2$ there is a constant $C(\alpha_0)$ such that \begin{gather*} \left|f_\alpha(x)-f_2(x)\right| \leq C(\alpha_0)(1+|x|)\|h\|_\infty d^{3/2}(2-\alpha)\log\frac{1}{2-\alpha} \quad\text{for all $x$}. \end{gather*}

We apply a Lindeberg principle under the Markov process setting to approximate the Wright-Fisher model with neutral $r$-alleles using a diffusion process, deriving an error rate based on a function class distance involving fourth-order bounded differentiable functions. This error rate consists of a linear combination of the maximum mutation rate and the reciprocal of the population size. Our result improves the error bound in the seminal work [PNAS,1977], where only the special case $r=2$ was studied.

We shall use the Euler-Maruyama (EM) scheme with decreasing step size $\Lambda=(\eta_n)_{n\in \mathbb{N}}$ to approximate the steady state for piecewise $\alpha$-stable Ornstein-Uhlenbeck processes arising in queue networks. These processes do not have an explicit dissipation. We prove the EM scheme converges to the steady state with a rate $\eta^{1/\alpha}_n$ in Wasserstein-1 distance. In addition, we utilize the Sinkhorn--Knopp algorithm to compute the Wasserstein-1 distance and conduct the simulations for several examples., Comment: 30 pages,7 figures

We study in this paper the EM scheme for a family of well-posed critical SDEs with the drift $-x\log(1+|x|)$ and $\alpha$-stable noises. Specifically, we find that when the SDE is driven by a rotationally symmetric $\alpha$-stable processes with $\alpha=2$ (i.e. Brownian motion), the EM scheme is bounded in the $L^2$ sense uniformly w.r.t. the time. In contrast, if the SDE is driven by a rotationally symmetric $\alpha$-stable process with $\alpha \in (0,2)$, all the $\beta$-th moments, with $\beta \in (0,\alpha)$, of the EM scheme blow up. This demonstrates a phase transition phenomenon as $\alpha \uparrow 2$. We verify our results by simulations.

Under certain general conditions, we prove that the stable central limit theorem holds in the total variation distance and get its optimal convergence rate for all $\alpha \in (0,2)$. Our method is by two measure decompositions, one step estimates, and a very delicate induction with respect to $\alpha$. One measure decomposition is light tailed and borrowed from \cite{BC16}, while the other one is heavy tailed and indispensable for lifting convergence rate for small $\alpha$. The proof is elementary and composed of the ingredients at the postgraduate level. Our result clarifies that when $\alpha=1$ and $X$ has a symmetric Pareto distribution, the optimal rate is $n^{-1}$ rather than $n^{-1} (\ln n)^2$ as conjectured in literatures.

Let $(X_t)_{t \ge 0}$ be the solution of the stochastic differential equation $$dX_t = b(X_t) dt+A dZ_t, \quad X_{0}=x,$$ where $b: \mathbb{R}^d \rightarrow \mathbb R^d$ is a Lipschitz function, $A \in \mathbb R^{d \times d}$ is a positive definite matrix, $(Z_t)_{t\geq 0}$ is a $d$-dimensional rotationally invariant $\alpha$-stable L\'evy process with $\alpha \in (1,2)$ and $x\in\mathbb{R}^{d}$. We use two Euler-Maruyama schemes with decreasing step sizes $\Gamma = (\gamma_n)_{n\in \mathbb{N}}$ to approximate the invariant measure of $(X_t)_{t \ge 0}$: one with i.i.d. $\alpha$-stable distributed random variables as its innovations and the other with i.i.d. Pareto distributed random variables as its innovations. We study the convergence rate of these two approximation schemes in the Wasserstein-1 distance. For the first scheme, when the function $b$ is Lipschitz and satisfies a certain dissipation condition, we show that the convergence rate is $\gamma^{1/\alpha}_n$. Under an additional assumption on the second order directional derivatives of $b$, this convergence rate can be improved to $\gamma^{1+\frac 1 {\alpha}-\frac{1}{\kappa}}_n$ for any $\kappa \in [1,\alpha)$. For the second scheme, when the function $b$ is twice continuously differentiable, we obtain a convergence rate of $\gamma^{\frac{2-\alpha}{\alpha}}_n$. We show that the rate $\gamma^{\frac{2-\alpha}{\alpha}}_n$ is optimal for the one dimensional stable Ornstein-Uhlenbeck process. Our theorems indicate that the recent remarkable result about the unadjusted Langevin algorithm with additive innovations can be extended to the SDEs driven by an $\alpha$-stable L\'evy process and the corresponding convergence rate has a similar behaviour. Compared with the previous result, we have relaxed the second order differentiability condition to the Lipschitz condition for the first scheme.

Consider the following stochastic differential equation for $(X_t)_{t\ge 0}$ on $\mathbb R^d$ and its Euler-Maruyama (EM) approximation $(Y_{t_n})_{n\in \mathbb Z^+}$: \begin{align*} &d X_t=b( X_t) d t+\sigma(X_t) d B_t, \\ & Y_{t_{n+1}}=Y_{t_{n}}+\eta_{n+1} b(Y_{t_{n}})+\sigma(Y_{t_{n}})\left(B_{t_{n+1}}-B_{t_{n}}\right), \end{align*} where $b:\mathbb{R}^d \rightarrow \mathbb{R}^d,\ \ \sigma: \mathbb R^d \rightarrow \mathbb{R}^{d \times d}$ are measurable, $B_t$ is the $d$-dimensional Brownian motion, $t_0:=0,t_{n}:=\sum_{k=1}^{n} \eta_{k}$ for constants $\eta_k>0$ satisfying $\lim_{k \rightarrow \infty} \eta_k=0$ and $\sum_{k=1}^\infty\eta_k =\infty$. Under (partial) dissipation conditions ensuring the ergodicity, we obtain explicit convergence rates of $\mathbb W_p(\mathscr{L}(Y_{t_n}), \mathscr{L}(X_{t_n}))+\mathbb W_p(\mathscr{L}(Y_{t_n}), \mu)\rightarrow 0$ as $n\rightarrow \infty$, where $\mathbb W_p$ is the $L^p$-Wasserstein distance for certain $p\in [0,\infty)$, $\mathscr{L}(\xi)$ is the distribution of random variable $\xi$, and $\mu$ is the unique invariant probability measure of $(X_t)_{t \ge 0}$. Comparing with the existing results where $b$ is at least $C^2$-smooth, our estimates apply to Hoelder continuous drift and can be sharp in several specific situations.

We are interested in the following two $\mathbb{R}^d$-valued stochastic differential equations (SDEs): \begin{gather*} d X_t=b(X_t)\,d t + \sigma\,d L_t, \quad X_0=x, %\label{BM-SDE} d Y_t=b(Y_t)\,d t + \sigma\,d B_t, \quad Y_0=y, \end{gather*} where $\sigma$ is an invertible $d\times d$ matrix, $L_t$ is a rotationally symmetric $\alpha$-stable L\'evy process, and $B_t$ is a $d$-dimensional standard Brownian motion (note that $B_t$ is a rotationally symmetric $\alpha$-stable L\'evy process with $\alpha=2$). We show that for any $\alpha_0 \in (1,2)$ the Wasserstein-$1$ distance $W_1$ satisfies for $\alpha \in [\alpha_0,2)$ \begin{gather*} W_{1}\left(X_{t}^x, Y_{t}^y\right) \leq C_1 e^{-C_2t}|x-y| +\frac{C}{\alpha_0-1}(2-\alpha)d\log(1+d), \end{gather*} which implies, in particular, \begin{equation} \label{e:W1Rate} W_1(\mu_\alpha, \mu_2) \leq \frac{C}{\alpha_0-1}(2-\alpha)d\log(1+d), \end{equation} where $\mu_\alpha$ and $\mu_2$ are the ergodic measures of $X_t$ and $Y_t$ respectively. For the special case of a $d$-dimensional Ornstein--Uhlenbeck system, we show that $W_1(\mu_\alpha, \mu_2) \geq C_{d} (2-\alpha)$ for all $\alpha\in(1,2)$; this indicates that the convergence rate with respect to $\alpha$ in the second bound is optimal. The term $d\log(1+d)$ appearing in this bound seems to be optimal for the dimension $d$ as well.

We study the approximation of the ergodic measure of the following stochastic differential equation (SDE) on $\mathbb{R}^d$: \begin{eqnarray}\label{e:SDEE} d X_t &=& (b_1(X_t)+b_2(X_t)) d t+\sigma(X_t) d W_t, \end{eqnarray} where $W_t$ is a $d$-dimensional standard Brownian motion, and $b_1: \mathbb{R}^d \mapsto \mathbb{R}^d$, $b_2: \mathbb{R}^d \mapsto \mathbb{R}^d$ and $\sigma: \mathbb{R}^d \mapsto \mathbb{R}^{d\times d}$ are the functions to be specified in Assumption 2.1 below. In particular, $b_1$ satisfies $b_1\in \mathbb{L}^\infty(\mathbb{R}^d)\cap \mathbb{L}^1(\mathbb{R}^d)$ or $b_1 \in \mathcal{C}_b^{\alpha}(\mathbb{R}^d)$ with $\alpha\in (0,1)$, which makes the standard numerical schemes not work or fail to give a good convergence rate. In order to overcome these two difficulties, we first apply a Zvonkin's transform to SDE and obtain a new SDE which has coefficients with nice properties and admits a unique ergodic measure $\widehat \mu$, then discretize the new equation by Euler-Maruyama scheme to approximate $\widehat \mu$, and finally use the inverse Zvonkin's transform to get an approximation of the ergodic measure of SDE, denoted by $\mu$. Our approximation method is inspired by Xie and Zhang [22]. The proof of our main result is based on the method of introducing a stationary Markov chain, a key ingredient in this method is establishing the regularity of a Poisson equation, which is done by combining the classical PDE local regularity and a nice extension trick introduced by Gurvich [10].

This paper studies the distribution estimation of contaminated data by the MoM-GAN method, which combines generative adversarial net (GAN) and median-of-mean (MoM) estimation. We use a deep neural network (DNN) with a ReLU activation function to model the generator and discriminator of the GAN. Theoretically, we derive a non-asymptotic error bound for the DNN-based MoM-GAN estimator measured by integral probability metrics with the $b$-smoothness H\"{o}lder class. The error bound decreases essentially as $n^{-b/p}\vee n^{-1/2}$, where $n$ and $p$ are the sample size and the dimension of input data. We give an algorithm for the MoM-GAN method and implement it through two real applications. The numerical results show that the MoM-GAN outperforms other competitive methods when dealing with contaminated data.

The generative adversarial networks (GANs) have recently been applied to estimating the distribution of independent and identically distributed data, and have attracted a lot of research attention. In this paper, we use the blocking technique to demonstrate the effectiveness of GANs for estimating the distribution of stationary time series. Theoretically, we derive a non-asymptotic error bound for the Deep Neural Network (DNN)-based GANs estimator for the stationary distribution of the time series. Based on our theoretical analysis, we propose an algorithm for estimating the change point in time series distribution. The two main results are verified by two Monte Carlo experiments respectively, one is to estimate the joint stationary distribution of $5$-tuple samples of a 20 dimensional AR(3) model, the other is about estimating the change point at the combination of two different stationary time series. A real world empirical application to the human activity recognition dataset highlights the potential of the proposed methods.

Inspired by \citet{Berkes14} and \citet{Wu07}, we prove an almost sure invariance principle for stationary $\beta-$mixing stochastic processes defined on Hilbert space. Our result can be applied to Markov chain satisfying Meyn-Tweedie type Lyapunov condition and thus generalises the contraction condition in \citet[Example 2.2]{Berkes14}. We prove our main theorem by the big and small blocks technique and an embedding result in \citet{gotze2011estimates}. Our result is further applied to the ergodic Markov chain and functional autoregressive processes.

In this paper, we develop a stochastic algorithm based on the Euler--Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of $G/Ph/n+GI$ queues in the Halfin-Whitt regime. Specifically, we prove a non-asymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion. To establish the error bound, we employ the recently developed Stein's method for multi-dimensional diffusions, in which the regularity of Stein's equation developed by Gurvich (2014, 2022) plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of $G/Ph/n+GI$ queues and its Euler-Maruyama scheme. In particular, the variances in the CLT and MDP associated with the limiting diffusion are determined by Stein's equation and Malliavin calculus, in which properties of a mollified diffusion and an associated weighted occupation time play a crucial role., Comment: arXiv admin note: substantial text overlap with arXiv:2109.03623

In this paper, we establish normalized and self-normalized Cram\'er-type moderate deviations for Euler-Maruyama scheme for SDE. As a consequence of our results, Berry-Esseen's bounds and moderate deviation principles are also obtained. Our normalized Cram\'er-type moderate deviations refines the recent work of [Lu, J., Tan, Y., Xu, L., 2022. Central limit theorem and self-normalized Cram\'er-type moderate deviation for Euler-Maruyama scheme. Bernoulli 28(2): 937--964]., Comment: 21 pages

Letting~$N=\left\{N(t), t\geq0\right\}$ be a standard Poisson process, Stroock~ \cite{Stroock-1981} constructed a family of continuous processes by $$\Theta_{\epsilon}(t)=\int_0^t\theta_{\epsilon}(r)dr, \ \ \ \ \ 0 \le t \le 1,$$ where $\theta_{\epsilon}(r)=\frac{1}{\epsilon}(-1)^{N(\epsilon^{-2}r)}$, and proved that it weakly converges to a standard Brownian motion under the continuous function topology. We establish the functional large deviations principle (LDP) for the approximations of a class of Gaussian processes constructed by integrals over $\Theta_{\epsilon}(t)$, and find the explicit form for rate function. As an application, we consider the following (non-Markovian) stochastic differential equation \begin{equation*} \begin{aligned} X^{\epsilon}(t) &=x_{0}+\int^{t}_{0}b(X^{\epsilon}(s))ds+\lambda(\epsilon)\int^{t}_{0}\sigma(X^{\epsilon}(s))d\Theta_{\epsilon}(s), \end{aligned} \end{equation*} where $b$ and $\sigma$ are both Lipschitz functions, and establish its Freidlin-Wentzell type LDP as $\epsilon \rightarrow 0$. The rate function indicates a phase transition phenomenon as $\lambda(\epsilon)$ moves from one region to the other.

We consider a stochastic 2D Navier-Stokes equation in a bounded domain. The random force is assumed to be non-degenerate and periodic in time, its law has a support localised with respect to both time and space. Slightly strengthening the conditions in the pioneering work about exponential ergodicity by Shirikyan [Shi15], we prove that the stochastic system satisfies Donsker-Varadhan typle large deviations principle. Our proof is based on a criterion of [JNPS15] in which we need to verify uniform irreducibility and uniform Feller property for the related Feynman-Kac semigroup.

We propose two Euler-Maruyama (EM) type numerical schemes in order to approximate the invariant measure of a stochastic differential equation (SDE) driven by an $\alpha$-stable L\'evy process ($1<\alpha<2$): an approximation scheme with the $\alpha$-stable distributed noise and a further scheme with Pareto-distributed noise. Using a discrete version of Duhamel's principle and Bismut's formula in Malliavin calculus, we prove that the error bounds in Wasserstein-$1$ distance are in the order of $\eta^{1-\epsilon}$ and $\eta^{\frac2{\alpha}-1}$, respectively, where $\epsilon \in (0,1)$ is arbitrary and $\eta$ is the step size of the approximation schemes. For the Pareto-driven scheme, an explicit calculation for Ornstein--Uhlenbeck $\alpha$-stable process shows that the rate $\eta^{\frac2{\alpha}-1}$ cannot be improved., Comment: Accepted by SPA. This version is a little different version from the published one, in which we add in the appendix a shorter proof for a Bismut-Elworthy-Li formula suggested by a very careful and knowledgeable referee

The purpose of this paper is to establish the Donsker-Varadhan type large deviations principle (LDP) for the two-dimensional stochastic Navier-Stokes system. The main novelty is that the noise is assumed to be highly degenerate in the Fourier space. The proof is carried out by using a criterion for the LDP developed in arXiv:1410.6188 in a discrete-time setting and extended in arXiv:1505.03686 to the continuous-time. One of the main conditions of that criterion is the uniform Feller property for the Feynman-Kac semigroup, which we verify by using Malliavin calculus.

There has been a surge of interest in developing robust estimators for models with heavy-tailed and bounded variance data in statistics and machine learning, while few works impose unbounded variance. This paper proposes two type of robust estimators, the ridge log-truncated M-estimator and the elastic net log-truncated M-estimator. The first estimator is applied to convex regressions such as quantile regression and generalized linear models, while the other one is applied to high dimensional non-convex learning problems such as regressions via deep neural networks. Simulations and real data analysis demonstrate the {robustness} of log-truncated estimations over standard estimations., Comment: 44 pages

In this paper, we develop a stochastic algorithm based on Euler-Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of the $M/Ph/n+M$ queue. Specifically, we prove a non-asymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion of the queueing model. Our result also provides an approximation to the steady-state of the diffusion-scaled queueing processes in the Halfin-Whitt regime given the well established interchange of limits property. To establish the error bound, we employ the recently developed Stein's method for multi-dimensional diffusions, in which the regularity of Stein's equation developed by Gurvich \cite{Gur1} plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of the $M/Ph/n+M$ queue and its Euler-Maruyama scheme. In particular, the variance of the CLT of the limiting queue is determined by using Stein's equation and Malliavin calculus., Comment: We updated the regularity estimate of Stein's equation and added CLT and LDP about the ergodic measures. The convergence rate of the approximation is improved

We study in this paper a weak approximation to stochastic variance reduced gradient Langevin dynamics by stochastic delay differential equations in Wasserstein-1 distance, and obtain a uniform error bound. Our approach is via a refined Lindeberg principle and Malliavin calculus., Comment: We update the assumption and the convergence rate, which are both better than the original version

We consider a stochastic differential equation and its Euler-Maruyama (EM) scheme, under some appropriate conditions, they both admit a unique invariant measure, denoted by $\pi$ and $\pi_\eta$ respectively ($\eta$ is the step size of the EM scheme). We construct an empirical measure $\Pi_\eta$ of the EM scheme as a statistic of $\pi_\eta$, and use Stein's method developed in \citet{FSX19} to prove a central limit theorem of $\Pi_\eta$. The proof of the self-normalized Cram\'er-type moderate deviation (SNCMD) is based on a standard decomposition on Markov chain, splitting $\eta^{-1/2}(\Pi_\eta(.)-\pi(.))$ into a martingale difference series sum $\mcl H_\eta$ and a negligible remainder $\mcl R_\eta$. We handle $\mcl H_\eta$ by the time-change technique for martingale, while prove that $\mcl R_\eta$ is exponentially negligible by concentration inequalities, which have their independent interest. Moreover, we show that SNCMD holds for $x = o(\eta^{-1/6})$, which has the same order as that of the classical result in \citet{shao1999cramer,JSW03}., Comment: To appear in Bernoulli, correct a small error in the proof of Lemma 4.1

We view the classical Lindeberg principle in a Markov process setting to establish a probability approximation framework by the associated It\^{o}'s formula and Markov operator. As applications, we study the error bounds of the following three approximations: approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by multiplicative Brownian motion, Euler-Maruyama (EM) discretization for multi-dimensional Ornstein-Uhlenbeck stable process, and multivariate normal approximation. All these error bounds are in Wasserstein-1 distance., Comment: Accepted by Annals of Applied Probability

We generalize the { ${\rm M}$-estimator} put forward by Catoni in his seminal paper [C12] to the case in which samples can have finite $\alpha$-th moment with $\alpha \in (1,2)$ rather than finite variance, our approach is by slightly modifying the influence function $\varphi$ therein. The choice of the new influence function is inspired by the Taylor-like expansion developed in [C-N-X]. We obtain a deviation bound of the estimator, as $\alpha \rightarrow 2$, this bound is the same as that in [C12]. Experiment shows that our generalized ${\rm M}$-estimator performs better than the empirical mean estimator, the smaller the $\alpha$ is, the better the performance will be. As an application, we study an $\ell_{1}$ regression considered by Zhang et al. [Z-Z] who assumed that samples have finite variance, and relax their assumption to be finite {$\alpha$-th} moment with $\alpha \in (1,2)$.

We study stochastic integrals driven by a general subordinator and establish a zero-one law for the finiteness of the resulting integral as well as moment estimates. As an application, we use these results to obtain structural properties of SPDEs driven by multiplicative pure jump noise, which include (1) a maximal inequality for a multiplicative stochastic convolution $Z_t$, (2) a small ball probability of $Z_t$, (3) the existence of invariant measures and accessibility to zero of SPDEs, and (4) a Galerkin approximation of solutions to SPDEs., Comment: Accepted by TAMS

We prove a large deviation principle (LDP) and a fluctuation theorem (FT) for the entropy production rate (EPR) of the following $d$ dimensional stochastic differential equation \begin{equation*} d X_{t}=AX_{t} d t+\sqrt{Q} d B_{t} \end{equation*} where $A$ is a real normal stable matrix, $Q$ is positive definite, and the matrices $A$ and $Q$ commute. The rate function for the EPR takes the following explicit form: \begin{equation*} I(x)=\left\{ \begin{array}{ll} x\frac{\sqrt{1+\ell_0(x)}-1}{2}+\frac 12\sum\limits_{k=1}^{d} \left(\sqrt{\alpha_k^2- \beta_k^2\ell_0(x)}+\alpha_k\right) , & x\ge 0 , \\ -x\frac{\sqrt{1+\ell_0(x)}+1}{2} +\frac 12\sum\limits_{k=1}^{d} \left(\sqrt{\alpha_k^2- \beta_k^2\ell_0(x)}+\alpha_k\right) , &x<0, \end{array} \right. \end{equation*} where $\alpha_{k}\pm {\rm i} \beta_{k}$ are the eigenvalues of $A$, and $\ell_0(x)$ is the unique solution of the equation: \begin{align*} |x|={\sqrt{1+\ell}} \times \sum_{k=1}^{d} \frac{\beta_k^2}{\sqrt{\alpha_k^2 -\ell\beta_k^2} },\qquad -1 \le \ell< \min_{k=1,...,d}\{\frac{\alpha_k^2}{\beta_k^2}\}. \end{align*} Simple closed form formulas for rate functions are rare and our work identifies an important class of large deviation problems where such formulas are available. The logarithmic moment generating function (the fluctuation function) $\Lambda$ associated with the LDP has a closed form (see it in the paper). The functions $\Lambda(\lambda)$ and $ I(x)$ satisfy the Cohen-Gallavotti symmetry properties. In particular, the functions $I$ and $\Lambda$ do not depend on the diffusion matrix $Q$, and are determined completely by the real and imaginary parts of the eigenvalues of $A$. Formally, the deterministic system with $Q=0$ has zero EPR and thus the model exhibits a phase transition in that the EPR changes discontinuously at $Q = 0$., Comment: There are two reasons for replace the papers: 1. the title of the paper makes others misunderstand that the work is just a straightforward generalization of our work before; 2. the content in introduction is not comprehensive and does not indicate our main contributions, we have to make some big revisions to make this paper more readable

We consider a directed polymer model in dimension $1+1$, where the disorder is given by the occupation field of a Poisson system of independent random walks on $\mathbb Z$. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation (SHE) with a Gaussian noise, whose space-time covariance is given by the heat kernel. In contrast to the case with space-time white noise where the solution of the SHE admits a Wiener-It\^{o} chaos expansion, we establish an $L^1$-convergent chaos expansions of iterated integrals generated by Picard iterations. Using this expansion and its discrete counterpart for the polymer partition functions, the convergence of the terms in the expansion is proved via functional analytic arguments and heat kernel estimates. The Poisson random walk system is amenable to careful moment analysis, which is an important input to our arguments., Comment: 45 pages. To appear in Journal of Functional Analysis

By a delicate analysis for the Stein's equation associated to the $\alpha$-stable law approximation with $\alpha \in (0,2)$, we prove a quantitative stable central limit theorem in Wasserstein type distance, which generalizes the results in the series of work \cite{Xu19, CNX19+,CNXYZ19+} from the univariate case to the multiple variate case. From an explicit computation for Pareto's distribution, we see that the rate of our approximation is sharp. The analysis of the Stein's equation is new and has independent interest.

We develop Stein's method for $\alpha$-stable approximation with $\alpha\in(0,1]$, continuing the recent line of research by Xu \cite{lihu} and Chen, Nourdin and Xu \cite{C-N-X} in the case $\alpha\in(1,2).$ The main results include an intrinsic upper bound for the error of the approximation in a variant of Wasserstein distance that involves the characterizing differential operators for stable distributions, and an application to the generalized central limit theorem. Due to the lack of first moment for the approximating sequence in the latter result, we appeal to an additional truncation procedure and investigate fine regularity properties of the solution to Stein's equation.

A Freidlin-Wentzell type large deviation principle is established for stochastic partial differential equations with slow and fast time-scales, where the slow component is a one-dimensional stochastic Burgers equation with small noise and the fast component is a stochastic reaction-diffusion equation. Our approach is via the weak convergence criterion developed in [3]., Comment: 30 pages

By the Lindeberg principle, we develop in this paper an approximation to one dimensional (possibly) asymmetric $\alpha$-stable distributions with $\alpha \in (0,2)$ in the smooth Wasserstein distance. It is the first time that the general stable central limit theorem is proved by the Lindeberg principle, and that this theorem with $\alpha \in (0,1]$ is proved by a new method other than Fourier analysis. Our main tools are a Taylor-like expansion and a Kolmogorov forward equation.

This paper is concerned with the Stein's method associated with a (possibly) asymmetric $\alpha$-stable distribution $Z$, in dimension one. More precisely, its goal is twofold. In the first part, we exhibit a genuine bound for the Wasserstein distance between $Z$ and any integrable random variable $X$, in terms of an operator that reduces to the classical fractional Laplacian in the symmetric case. Then, in the second part we apply the aforementioned bound to compute error rates in the stable central limit theorem, when the entries are in the domain $\mathcal{D}_\alpha$ of normal attraction of a stable law of exponent $\alpha$. To conclude, we study the specific case where the entries are Pareto like multiplied by a slowly varying function, which provides an example of random variables that do not belong to $\mathcal{D}_\alpha$, but for which our approach continues to apply., Comment: 14 pages. A new example where the entries are not in the domain of normal attraction of a stable law of exponent $\alpha$ has been added

It is well known that $\int_{0}^{1} t^{-\theta} d t<\infty$ for $\theta \in (0,1)$ and $\int_{0}^{1} t^{-\theta} d t=\infty$ for $\theta \in [1,\infty)$. Since $t$ can be taken as an $\alpha$-stable subordinator with $\alpha=1$, it is natural to ask whether $\int_{0}^{1} t^{-\theta} d S_{t}$ has a similar property when $S_{t}$ is an $\alpha$-stable subordinator with $\alpha \in (0,1)$. We show that $\theta=\frac 1\alpha$ is the border line such that $\int_{0}^{1} t^{-\theta} d S_{t}$ is finite a.s. for $\theta \in (0, \frac 1\alpha)$ and blows up a.s. for $\theta \in [\frac1\alpha,\infty)$. When $\alpha=1$, our result recovers that of $\int_{0}^{1} t^{-\theta} d t$. Moreover, we give a $p$-th moment estimate for the integral when $\theta \in (0,\frac 1\alpha)$., Comment: 5 page

Stein's method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second- or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismut's formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein's exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wasserstein distance.We apply the theorem to the unadjusted Langevin algorithm., Comment: We changed the title and corrected some small errors and typos

Let $n \in \mathbb N$, let $\zeta_{n,1},...,\zeta_{n,n}$ be a sequence of independent random variables with $\mathbb E \zeta_{n,i}=0$ and $\mathbb E |\zeta_{n,i}|<\infty$ for each $i$, and let $\mu$ be an $\alpha$-stable distribution having characteristic function $e^{-|\lambda|^{\alpha}}$ with $\alpha\in (1,2)$. Denote $S_{n}=\zeta_{n,1}+...+\zeta_{n,n}$ and its distribution by $\mathcal L(S_n)$, we bound the Wasserstein distance of $\mathcal L(S_{n})$ and $\mu$ essentially by an $L^{1}$ discrepancy between two kernels, this bound can be interpreted as a generalization of the Stein discrepancy (in $L^{2}$ sense) introduced by Ledoux, Nourdin and Peccati. More precisely, we prove the following inequality: \begin{equation} \begin{split} d_W\left(\mathcal L (S_n), \mu\right) \ \le C \left[\sum_{i=1}^n\int_{-N}^N \left|\frac{\mathcal K_\alpha(t,N)}n -\frac{ K_i(t,N)}{\alpha}\right| d t \ +\ \mathcal R_{N,n}\right], \end{split} \end{equation} where $d_{W}$ is the Wasserstein distance of probability measures, $\mathcal K_\alpha(t,N)$ is the kernel of a decomposition of the fractional Laplacian $\Delta^{\frac \alpha2}$, $ K_i(t,N)$ is a kernel introduced by Chen, Goldstein and Shao with a truncation which can be interpreted as an $L^1$ Stein kernel, and $\mathcal R_{N,n}$ is a small remainder. The integral term $$\sum_{i=1}^n\int_{-N}^N \left|\frac{\mathcal K_\alpha(t,N)}n -\frac{ K_i(t,N)}{\alpha}\right| d t$$ can be interpreted as an $L^{1}$ Stein discrepancy. As an application, we prove a general theorem of stable law convergence rate when $\zeta_{n,i}$ are i.i.d. and the distribution falls in the normal domain of attraction of $\mu$. We also study four examples with comparing our convergence rates and those known for these examples, among which the distribution in the second example is not in the normal domain of attraction of $\mu$., Comment: We corrected some small typos. Accepted by Annals of Applied Probability

In this paper we prove uniform Hausdorff and packing dimension results for the images of a large family of Markov processes. The main tools are the two covering principles of Xiao (second author). As applications, uniform Hausdorff and packing dimension results for certain classes of L\'evy processes, stable jump diffusion and non-symmetric stable-like processes are obtained., Comment: We revised the assumptions (A2) and (A3) in the previous version, the new assumptions are more general and expected to cover more examples

We proposed a new penalized method in this paper to solve sparse Poisson Regression problems. Being different from $\ell_1$ penalized log-likelihood estimation, our new method can be viewed as penalized weighted score function method. We show that under mild conditions, our estimator is $\ell_1$ consistent and the tuning parameter can be pre-specified, which shares the same good property of the square-root Lasso.

We study the ergodicity of stochastic reaction-diffusion equation driven by subordinate Brownian motions. After establishing the strong Feller property and irreducibility of the system, we prove the tightness of the solution's law. These properties imply that this stochastic system admits a unique invariant measure according to Doob's and Krylov-Bogolyubov's theories. Furthermore, we establish a large deviation principle for the occupation measure of this system by a hyper-exponential recurrence criterion. It is well known that S(P)DEs driven by $\alpha$-stable type noises do not satisfy Freidlin-Wentzell type large deviation, our result gives an example that strong dissipation overcomes heavy tailed noises to produce a Donsker-Varadhan type large deviation as time tends to infinity., Comment: We added more references. arXiv admin note: text overlap with arXiv:1510.03522, arXiv:1601.06270

Using heat kernel estimates, we prove the pathwise uniqueness for strong solutions of irregular stochastic differential equation driven by a family of Markov process, whose generator is a non-local and non-symmetric L\'evy type operator. Due to the extra term $1_{[0,\sigma(X_{s-},z)]}(r)$ in multiplicative noise, we need to derive some new regularity results for the generator and use a trick of mixing $L_1$ and $L_2$-estimates by Kurtz and Protter \cite{Ku-Po}.

Under some regularity conditions on $b$, $\sigma$ and $\alpha$, we prove that the following perturbed stochastic differential equation \begin{equation} X_t=x+\int_0^t b(X_s)ds+\int_0^t \sigma(X_s) dB_s+\alpha \sup_{0 \le s \le t} X_s, \ \ \ \alpha<1 \end{equation} admits smooth densities for all $0 \le t \le t_0$, where $t_0>0$ is some finite number.

Using the hyper-exponential recurrence criterion, a large deviation principle for the occupation measure is derived for a class of non-linear monotone stochastic partial differential equations. The main results are applied to many concrete SPDEs such as stochastic $p$-Laplace equation, stochastic porous medium equation, stochastic fast-diffusion equation, and even stochastic real Ginzburg-Landau equation driven by $\alpha$-stable noises., Comment: This paper generalizes the idea in our NOT published paper arXiv:1510.03522. There is a substantial overlap with arXiv:1510.03522