Following Csörgő, Szyszkowicz and Wang (Ann. Statist. {\bf 34}, (2006), 1013--1044) we consider a long range dependent linear sequence. We prove weak convergence of the uniform Vervaat and the uniform Vervaat error processes, extending their results to distributions with unbounded support and removing normality assumption.
In this work we prove an asymptotic result, that under some conditions on the involved distribution functions, is valid for any Oppenheim expansion, extending a classical result proven by W. Vervaat in 1972 for denominators of the Luroth case. Furthermore, we study the convergence in distribution of weighted sums of a sequence of independent random variables. Although the result is of its own interest, in the present setting it is used to prove convergence in distribution of specific sequences of random variables generalizing known results obtained for Luroth random variables.
In this paper, we investigate the weak convergence rate of Euler-Maruyama's approximation for stochastic differential equations with irregular drifts. Explicit weak convergence rates are presented if drifts satisfy an integrability condition including discontinuous functions which can be non-piecewise continuous or in fractional Sobolev space.
The purpose of this paper is to prove a weak convergence result for empirical processes indexed in general classes of functions and with an underlying $α$-mixing sequence of random variables. In particular the uniformly boundedness assumption on the function class, which is required in most of the existing literature, is spared. Furthermore under strict stationarity a weak convergence result for the sequential empirical process indexed in function classes is obtained, as a direct consequence. Two examples in mathematical statistics, that cannot be treated with existing results, are given as po…
Our aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale related to the state indicator function, we provide sufficient conditions for weak convergence of stochastic integrals with re…
A common statistical task lies in showing asymptotic normality of certain statistics. In many of these situations, classical textbook results on weak convergence theory suffice for the problem at hand. However, there are quite some scenarios where stronger results are needed in order to establish an asymptotic normal approximation uniformly over a family of probability measures. In this note we collect some results in this direction. We restrict ourselves to weak convergence in $\mathbb R^d$ with continuous limit measures.
In this paper, by using Girsanov's transformation and the property of the corresponding reference stochastic differential equations, we investigate weak existence and uniqueness of solutions and weak convergence of Euler-Maruyama scheme to stochastic functional differential equations with Hölder continuous drift driven by fractional Brownian motion with Hurst index $H\in (1/2,1)$.
In this paper, we study the averaging principle for a class of stochastic differential equations driven by $α$-stable processes with slow and fast time-scales, where $α\in(1,2)$. We prove that the strong and weak convergence order are $1-1/α$ and $1$ respectively. We show, by a simple example, that $1-1/α$ is the optimal strong convergence rate.
In this paper we develop via Girsanov's transformation a perturbation argument to investigate weak convergence of Euler-Maruyama (EM) scheme for path-dependent SDEs with Hölder continuous drifts. This approach is available to other scenarios, e.g., truncated EM schemes for non-degenerate SDEs with finite memory or infinite memory. Also, such trick can be applied to study weak convergence of truncated EM scheme for a range of stochastic Hamiltonian systems with irregular coefficients and with memory, which are typical degenerate dynamical systems. Moreover, the weak convergence of path-dependen…
In this paper, we study the weak convergence of the extremes of supercritical branching Lévy processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, $\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of $\mathbb{X}_t$.
This work establishes the weak convergence of Euler-Maruyama's approximation for stochastic differential equations (SDEs) with singular drifts under the integrability condition in lieu of the widely used growth condition. This method is based on a skillful application of the dimension-free Harnack inequality. Moreover, when the drifts satisfy certain regularity conditions, the convergence rate is estimated. This method is also applicable when the diffusion coefficients are degenerate. A stochastic damping Hamiltonian system is studied as an illustrative example.
In this paper, we study the averaging principle for distribution dependent stochastic differential equations with drift in localized $L^p$ spaces. Using Zvonkin's transformation and estimates for solutions to Kolmogorov equations, we prove that the solutions of the original system strongly and weakly converge to the solution of the averaged system as the time scale $\eps$ goes to zero. Moreover, we obtain rates of the strong and weak convergence that depend on $p$ respectively.
We consider SDEs with bounded and $α$-Hölder continuous drift, with $α\in (0,1)$, driven by multiplicative noise. We show that under sufficient conditions on the diffusion matrix, which guarantee the existence of a unique strong solution, the weak rate of convergence for the Euler-Maruyama scheme is almost $(1+α)/2$. The present paper forms part of the author's master's thesis.
In this paper, we consider a fundamental class of stochastic differential equations with time delays. Our aim is to investigate the weak convergence with respect to delay parameter of the solutions. Based on the techniques of Malliavin calculus, we obtain an explicit estimate for the rate of convergence. An application to the Carathéodory approximation scheme of stochastic differential equations is provided as well.
Strong and weak approximation errors of a spatial finite element method are analyzed for stochastic partial differential equations(SPDEs) with one-sided Lipschitz coefficients, including the stochastic Allen--Cahn equation, driven by additive noise. In order to give the strong convergence rate of the finite element method, we present an appropriate decomposition and some a priori estimates of the discrete stochastic convolution. To the best of our knowledge, there has been no essentially sharp weak convergence rate of spatial approximation for parabolic SPDEs with non-globally Lipschitz coeffi…
For a strictly stationary sequence of random variables we derive functional convergence of the joint partial sum and partial maxima process under joint regular variation with index $α\in (0,2)$ and weak dependence conditions. The limiting process consists of an $α$--stable Lévy process and an extremal process. We also describe the dependence between these two components of the limit. The convergence takes place in the space of $\mathbb{R}^{2}$--valued càdlàg functions on $[0,1]$, with the Skorohod weak $M_{1}$ topology. We further show that this topology in general can not be replaced by the s…
For moving average processes with random coefficients and heavy-tailed innovations that are weakly dependent in the sense of strong mixing and local dependence condition $D'$ we study joint functional convergence of partial sums and maxima. Under the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we derive a functional limit theorem in the space of $\mathbb{R}^{2}$-valued càdlàg functions on $[0, 1]$ with the Skorokhod weak $M_{2}$ topology.
For solutions $X=(X_t)_{t\in[0,T]}$ of Lévy-driven Marcus stochastic differential equations we study the Wong--Zakai type time discrete approximations $\bar X=(\bar X_{kh})_{0\leq k\leq T/h}$, $h>0$, and establish the first order convergence $|E f(X_T)-E f(X^h_T)|\leq C h$ for $f\in C_b^4$.
(English) This monograph aims at presenting the core weak convergence theory for sequences of random vectors with values in $\mathbb{R}^k$. In some places, a more general formulation in metric spaces is provided. It lays out the necessary foundation that paves the way to applications in particular subfields of the theory. In particular, the needs of Asymptotic Statistics are addressed. A whole chapter is devoted to weak convergence in $\mathbb{R}$ where specific tools, for example for handling weak convergence of sequences using independent and indentically distributed random variables such th…
Using Zvonkin's transform and the Poisson equation in $R^d$ with a parameter, we prove the averaging principle for stochastic differential equations with time-dependent Hölder continuous coefficients. Sharp convergence rates with order $(α\wedge1)/2$ in the strong sense and $(α/2)\wedge1$ in the weak sense are obtained, considerably extending the existing results in the literature. Moreover, we prove that the convergence of the multi-scale system to the effective equation depends only on the regularity of the coefficients of the equation for the slow variable, and does not depend on the regula…
Necessary and sufficient conditions for weak and vague convergence of measures are important for a diverse host of applications. This paper aims to give a comprehensive description of the relationship between the two modes of convergence when the measures are signed, which is largely absent from the literature. Furthermore, when the underlying space is $\mathbb{R}$, we study the relationship between vague convergence of signed measures and the pointwise convergence of their distribution functions.
