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From the distributional characterizations that lie at the heart of Stein's method we derive explicit formulae for the mass functions of discrete probability laws that identify those distributions. These identities are applied to develop…

Methodology · Statistics 2022-02-16 Steffen Betsch , Bruno Ebner , Franz Nestmann

We combine Stein's method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main…

Probability · Mathematics 2008-11-19 Ivan Nourdin , Giovanni Peccati , Anthony Réveillac

We study the discrepancy between the distribution of a vector-valued functional of i.i.d. random elements and that of a Gaussian vector. Our main contribution is an explicit bound on the convex distance between the two distributions,…

Probability · Mathematics 2022-03-25 Mikołaj J. Kasprzak , Giovanni Peccati

This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form $X_k =…

Probability · Mathematics 2012-07-13 Mohamed El Machkouri , Dalibor Volny , Wei Biao Wu

We develop a new method for bounding the relative entropy of a random vector in terms of its Stein factors. Our approach is based on a novel representation for the score function of smoothly perturbed random variables, as well as on the de…

Probability · Mathematics 2013-08-20 Ivan Nourdin , Giovanni Peccati , Yvik Swan

For parameter estimation of continuous and discrete distributions, we propose a generalization of the method of moments (MM), where Stein identities are utilized for improved estimation performance. The construction of these Stein-type…

Methodology · Statistics 2023-12-25 Simon Nik , Christian H. Weiß

We study a novel class of affine invariant and consistent tests for multivariate normality. The tests are based on a characterization of the standard $d$-variate normal distribution by means of the unique solution of an initial value…

Statistics Theory · Mathematics 2020-07-07 Bruno Ebner , Norbert Henze , David Strieder

New lower bounds on the total variation distance between the distribution of a sum of independent Bernoulli random variables and the Poisson random variable (with the same mean) are derived via the Chen-Stein method. The new bounds rely on…

Information Theory · Computer Science 2013-07-17 Igal Sason

We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be written as products of independent random variables. We…

Probability · Mathematics 2018-09-28 Robert E. Gaunt , Guillaume Mijoule , Yvik Swan

Consider a graph on randomly scattered points in an arbitrary space, with two points $x,y$ connected with probability $\phi(x,y)$. Suppose the number of points is large but the mean number of isolated points is $O(1)$. We give general…

Probability · Mathematics 2017-09-21 Mathew D. Penrose

Stein operators allow to characterise probability distributions via differential operators. Based on these characterisations, we develop a new method of point estimation for marginal parameters of strictly stationary and ergodic processes,…

Statistics Theory · Mathematics 2024-12-05 Bruno Ebner , Adrian Fischer , Robert E. Gaunt , Babette Picker , Yvik Swan

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…

Probability · Mathematics 2019-04-16 Peng Chen , Ivan Nourdin , Lihu Xu , Xiaochuan Yang , Rui Zhang

We build on the formalism developed in [arXiv:1906.08372v1] to propose new representations of solutions to Stein equations. We provide new uniform and non uniform bounds on these solutions (a.k.a.\ Stein factors). We use these…

Probability · Mathematics 2019-11-14 Marie Ernst , Yvik Swan

The purpose of this paper is to synthesize the approaches taken by Chatterjee-Meckes and Reinert-R\"ollin in adapting Stein's method of exchangeable pairs for multivariate normal approximation. The more general linear regression condition…

Probability · Mathematics 2010-05-18 Elizabeth S. Meckes

This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean-covariance structure. This…

Statistics Theory · Mathematics 2024-04-04 Christian Genest , Frédéric Ouimet

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 compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein's method, as well as the use of appropriate discrete Malliavin operators. Although our approach does not…

Probability · Mathematics 2009-05-21 Ivan Nourdin , Giovanni Peccati , Gesine Reinert

Stein operators are differential operators which arise within the so-called Stein's method for stochastic approximation. We propose a new mechanism for constructing such operators for arbitrary (continuous or discrete) parametric…

Probability · Mathematics 2013-05-23 Christophe Ley , Yvik Swan

Distributional transformations characterized by equations relating expectations of test functions weighted by a given biasing function on the original distribution to expectations of the test function's higher derivatives with respect to…

Probability · Mathematics 2015-09-23 Christian Döbler

We study the distribution of the magnetization of the critical mean-field O(N) model with N > 1. Specifically, we bound the Wasserstein distance between the finite-volume and limiting distributions, in terms of the number of spins. To…

Probability · Mathematics 2025-12-08 Timothy M. Garoni , Aram Perez , Zongzheng Zhou
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