Stein's Method and Related Topics

Title: Local law of addition of random matrices.
Abstract: The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free additive convolution of their spectral distributions. In this talk, I will show that this convergence also holds locally, down to the optimal scales larger than the eigenvalue spacing.  This is a joint work with Laszlo Erdos and Kevin Schnelli.

Title: Stein's method for multivariate discrete normal approximation.
Abstract: Stein's method for distributional approximation is one of the most remarkable additions to the probabilist's toolkit in recent years. It can be used in a wide variety of settings, and in conjunction with any approximating distribution.  Two major achievements that have developed out of it are the Stein--Chen method for Poisson approximation in total variation, and Goetze's multivariate normal approximation with respect to the convex sets metric.  Here, we discuss recent work, related to both of these topics, on discrete multivariate normal approximation in total variation for integer valued random vectors. Joint work with Malwina Luczak and Aihua Xia.

Title: Stein's method and Palm theory in random measures.
Abstract: In this talk, I will discuss the connection between Stein’s method and Palm theory and show how this connection can be exploited to study Poisson process approximation for point processes and normal approximation for random measures, with applications to computational biology and stochastic geometry.

Title: Hitting time and mixing time bounds of Stein’s factors.
Abstract:  For any discrete target distribution, we exploit the connection between Markov chains and Stein's method via the generator approach and express the solution of Stein's equation in terms of expected hitting time. This yields new upper bounds of Stein's factors in terms of the parameters of the Markov chain, such as mixing time and the gradient of expected hitting time. We compare the performance of these bounds with those in the literature, and in particular we consider Stein's method for discrete uniform, binomial, geometric and hypergeometric distribution. As another application, the same methodology applies to bound expected hitting time via Stein's factors. This article highlights the interplay between Stein's method, modern Markov chain theory and classical fluctuation theory.

Title: Stein's Method for Steady-State Approximations of Queueing Systems.
Abstract: Through queueing systems modeling customer call centers and hospital patient flows, I will give an introduction on how to use Stein's method both as an engineering tool for generating good steady-state diffusion approximations and as a mathematical too for establishing error bounds of these approximations. These approximations are often universally accurate in multiple parameter regions, from underloaded, to critically loaded, to overloaded (when customers abandon).

Title: Higher order approximation for sequences converging in the mod-Gaussian sense.
Abstract: Recently, Barhoumi-Andreani connected the notion of mod-convergence with Stein’s method, developing and applying Stein’s method for certain penalised Gaussian distributions. We go on in this direction with applications for some dependence structures like the exchangeable pair approach as well as the theory of dependency graphs. This is joint work with two of my students, Carolin Kleemann and Marius Butzek.

Title: Stein's method for asymmetric alpha-stable distributions, with application to the stable CLT.
Abstract: This talk will be concerned with the Stein's method associated with a (possibly) asymmetric alpha-stable distribution in dimension one.Based on a recent collaboration with Peng Chen and Lihu, both from University of Macau.

Title: Branching random walk and Stein's method.
Abstract: For the critical nearest-neighbor multidimensional branching random walk conditional on non-extinction, we show convergence to an exponential distribution for the number of sites with a given multiplicity of particles.  We also get a rate of convergence using a version of Stein's method. We will also discuss connections to the multivariate Laplace and distribution.

Title: Central moment inequalities using Stein's method.
Abstract: We derive explicit central moment inequalities for random variables that admit a Stein coupling, such as exchangeable pairs, size-bias couplings or local dependence, among others. The bounds are in terms of moments (not necessarily central) of variables in the Stein coupling, which are typically positive or local in some sense, and therefore easier to bound. In cases where the Stein couplings have the kind of behaviour leading to good normal approximation, the central moments are closely bounded by those of a normal. We show how the bounds can be used to produce concentration inequalities, and compare to those existing in related settings. Finally, we illustrate the power of the theory by bounding the central moments of sums of neighbourhood statistics in sparse Erdős--Rényi random graphs. Joint work with A.D. Barbour and Yuting Wen.

Title: Cramér-type Moderate Deviation Theorems for Nonnormal Approximation.
Abstract: A Cramér-type moderate deviation theorem quantifies the relative error of the tail probability approximation. It provides theoretical justification when the limiting tail probability can be used to estimate the tail probability under study. Chen, Fang and Shao (2013) obtained a general Cramér-type moderate result using Stein's method when the limiting was a normal distribution. In this talk, we shall establish Cramér-type moderate deviation theorems for nonnormal approximation under a general Stein identity. Applications will also be discussed. This is based on a joint work with Mengchen Zhang and Zhuosong Zhang.

Title: On approximate distribution of the superposition of point processes.
Abstract: It is well known that the Poisson law of small numbers guarantees that a (compound) Poisson point process provides a good approximation to the superposition of sparse and "nearly" independent point processes. This also ensures that a suitable Poisson point process offers a good approximation to the superposition of independent point processes after most points have been thinned away and/or the "left-over" point processes are suitably rescaled. Inspired by a project in species distribution modelling, we consider approximate distribution of the superposition of independent point processes without thinning and scaling. In this talk, I'll mainly focus on a "modified" Poisson point process as the approximate distribution and establish a Berry-Esseen type theorem for the superposition of independent and identically distributed point processes. (This is a joint work with T Cong and F Zhang)

Title: Normal Approximation by Stein's Method under Sublinear Expectations.
Abstract: Peng (2007) proved the Central Limit Theorem under a sublinear expectation. In this talk, we shall give an estimate to the rate of convergence of this CLT by Stein's method under sublinear expectations.

Title: Berry-Esseen Bounds of Normal and Non-normal Approximation for Unbounded Exchangeable Pairs.
Abstract: An exchangeable pair approach is  commonly taken in the normal and non-normal approximation using Stein's method. It has been successfully used to identify the limiting distribution and provide an error of approximation. However, when the difference of the exchangeable pair is not bounded by a small deterministic constant, the error bound is often not optimal. In this paper, using the exchangeable pair approach of Stein's method,  a new Berry-Esseen bound for an arbitrary random variable is established  without a bound on the difference of the exchangeable pair.  We will also talk about the application to the general Curie-Weiss model. This is a joint work with Qi-Man Shao.