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We derive normal approximation bounds in the Wasserstein distance for sums of weighted U-statistics, based on a general distance bound for functionals of independent random variables of arbitrary distributions. Those bounds are applied to…

Probability · Mathematics 2020-07-28 Nicolas Privault , Grzegorz Serafin

Stein's method is used to approximate sums of discrete and locally dependent random variables by a centered and symmetric Binomial distribution. Under appropriate smoothness properties of the summands, the same order of accuracy as in the…

Probability · Mathematics 2007-05-23 Adrian Röllin

New bounds for the $k$-th order derivatives of the solutions of the normal and multivariate normal Stein equations are obtained. Our general order bounds involve fewer derivatives of the test function than those in the existing literature.…

Probability · Mathematics 2017-03-21 Robert E. Gaunt

We develop a functional Stein-Malliavin method in a non-diffusive Poissonian setting, thus obtaining a) quantitative central limit theorems for approximation of arbitrary non-degenerate Gaussian random elements taking values in a separable…

Probability · Mathematics 2023-04-17 Solesne Bourguin , Simon Campese , Thanh Dang

We use Stein's method to establish the rates of normal approximation in terms of the total variation distance for a large class of sums of score functions of marked Poisson point processes on $\mathbb{R}^d$. As in the study under the weaker…

Probability · Mathematics 2020-11-17 Tianshu Cong , Aihua Xia

We adapt Stein's method of diffusion approximations, developed by Barbour, to the study of chaotic dynamical systems. We establish an error bound in the functional central limit theorem with respect to an integral probability metric of…

Dynamical Systems · Mathematics 2025-11-05 Juho Leppänen , Yuto Nakajima , Yushi Nakano

Building on the rather large literature concerning the regularity of the solution of the standard normal Stein equation, we provide a complete description of the best possible uniform bounds for the derivatives of the solution of the…

Probability · Mathematics 2024-12-10 Robert E. Gaunt

Stein's method is used to study discrete representations of multidimensional distributions that arise as approximations of states of quantum harmonic oscillators. These representations model how quantum effects result from the interaction…

Probability · Mathematics 2021-05-31 Ian W. McKeague , Yvik Swan

In this paper we propose a new, simple and explicit mechanism allowing to derive Stein operators for random variables whose characteristic function satisfies a simple ODE. We apply this to study random variables which can be represented as…

Probability · Mathematics 2023-07-06 Benjamin Arras , Ehsan Azmoodeh , Guillaume Poly , Yvik Swan

In this paper, we obtain error bound for binomial and negative binomial approximations to weighted sums of locally dependent random variables, using Stein's method. We also discuss approximation results for weighted sums of independent…

Probability · Mathematics 2020-10-20 Amit N. Kumar

We introduce a new version of Stein's method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and concentration of measure. Unlike Skorokhod…

Probability · Mathematics 2009-09-29 Sourav Chatterjee

In this article, we consider Poisson and Poisson convoluted geometric approximation to the sums of $n$ independent random variables under moment conditions. We use Stein's method to derive the approximation results in total variation…

Probability · Mathematics 2020-07-07 Pratima Eknath Kadu

We propose a new functional analytic approach to Stein's method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property. We make use of this theory in order to derive abstract…

Probability · Mathematics 2020-08-13 Christian Döbler

In this paper we extend Stein's method to the distribution of the product of $n$ independent mean zero normal random variables. A Stein equation is obtained for this class of distributions, which reduces to the classical normal Stein…

Probability · Mathematics 2017-05-30 Robert E. Gaunt

We present an adaptation of Stein's method of normal approximation to the study of both discrete- and continuous-time dynamical systems. We obtain new correlation-decay conditions on dynamical systems for a multivariate central limit…

Probability · Mathematics 2017-01-12 Olli Hella , Juho Leppänen , Mikko Stenlund

We obtain a Stein characterisation of the distribution of the product of two correlated normal random variables with non-zero means, and more generally the distribution of the sum of independent copies of such random variables. Our Stein…

Probability · Mathematics 2024-09-11 Robert E. Gaunt , Siqi Li , Heather L. Sutcliffe

In this paper, we present a minimal formalism for Stein operators which leads to different probabilistic representations of solutions to Stein equations. These in turn provide a wide family of Stein-Covariance identities which we put to use…

Probability · Mathematics 2018-12-27 Marie Ernst , Gesine Reinert , Yvik Swan

This paper concerns the development of Stein's method for chi-square approximation and its application to problems in statistics. New bounds for the derivatives of the solution of the gamma Stein equation are obtained. These bounds involve…

Probability · Mathematics 2017-05-30 Robert E. Gaunt , Alastair Pickett , Gesine Reinert

In this paper, we consider the sums of non-negative integer valued $m$-dependent random variables, and its approximation to the power series distribution. We first discuss some relevant results for power series distribution such as Stein…

Probability · Mathematics 2020-05-05 Amit N. Kumar , Neelesh S. Upadhye , P. Vellaisamy

New bounds on the total variation distance between the law of integer valued functionals of possibly non-symmetric and non-homogeneous infinite Rademacher sequences and the Poisson distribution are established. They are based on a…

Probability · Mathematics 2017-07-26 Kai Krokowski