A note on the exchangeability condition in Stein's method

Rates in the Central Limit Theorem and diffusion approximation via Stein's Method

We present a way to use Stein's method in order to bound the Wasserstein distance of order $2$ between two measures $ν$ and $μ$ supported on $\mathbb{R}^d$ such that $μ$ is the reversible measure of a diffusion process. In order to apply our result, we only require to have access to a stochastic process $(X_t)_{t \geq 0}$ such that $X_t$ is drawn from $ν$ for any $t > 0$. We then show that, whenever $μ$ is the Gaussian measure $γ$, one can use a slightly different approach to bound the Wasserstein distances of order $p \geq 1$ between $ν$ and $γ$ under an additional exchangeability assumption …

两等位基因Moran模型的Beta近似：Stein方法

Beta approximation for the two alleles Moran model by Stein's method

本文针对两等位基因Moran模型，利用Stein方法给出Beta近似的精确误差项，并证明其符合Doebler框架。

A note on one-dimensional Poincaré inequalities by Stein-type integration

We study the weighted Poincaré constant $C(p,w)$ of a probability density $p$ with weight function $w$ using integration methods inspired by Stein's method. We obtain a new version of the Chen-Wang variational formula which, as a byproduct, yields simple upper and lower bounds on $C(p,w)$ in terms of the so-called Stein kernel of $p$. We also iterate these variational formulas so as to build sequences of nested intervals containing the Poincaré constant, sequences of functions converging to said constant, as well as sequences of functions converging to the solutions of the corresponding spectr…

关于随机图极值度分布的注记：基于Stein-Chen方法

A note on the distribution of the extreme degrees of a random graph via the Stein-Chen method

本文用Stein-Chen方法重新证明Bollobás关于随机图极值度分布的定理，并给出收敛速率，且方法可推广至边概率依赖顶点数的情形。

Limit theorems for invariant distributions

A distributional symmetry is invariance of a distribution under a group of transformations. Exchangeability and stationarity are examples. We explain that a result of ergodic theory provides a law of large numbers: If the group satisfies suitable conditions, expectations can be estimated by averaging over subsets of transformations, and these estimators are strongly consistent. We show that, if a mixing condition holds, the averages also satisfy a central limit theorem, a Berry-Esseen bound, and concentration. These are extended further to apply to triangular arrays, to randomly subsampled ave…

Stein operators for variables form the third and fourth Wiener chaoses

Let $Z$ be a standard normal random variable and let $H_n$ denote the $n$-th Hermite polynomial. In this note, we obtain Stein equations for the random variables $H_3(Z)$ and $H_4(Z)$, which represents a first step towards developing Stein's method for distributional limits from the third and fourth Wiener chaoses. Perhaps surprisingly, these Stein equations are fifth and third order linear ordinary differential equations, respectively. As a warm up, we obtain a Stein equation for the random variable $aZ^2+bZ+c$, $a,b,c\in\mathbb{R}$, which leads us to a Stein equation for the non-central chi-…

On the geometry of a class of invariant measures and a problem of Aldous

In his 1985 survey of notions of exchangeability, Aldous introduced a form of exchangeability corresponding to the symmetries of the infinite discrete cube, and asked whether these exchangeable probability measures enjoy a representation theorem similar to those for exchangeable sequences, arrays and set-indexed families. In this note we to prove that, whereas the known representation theorems for different classes of partially exchangeable probability measure imply that the compact convex set of such measures is a Bauer simplex (that is, its subset of extreme points is closed), in the case of…

Step Size in Stein's Method of Exchangeable Pairs

Stein's method of exchangeable pairs is examined through five examples in relation to Poisson and normal distribution approximation. In particular, in the case where the exchangeable pair is constructed from a reversible Markov chain, we analyze how modifying the step size of the chain in a natural way affects the error term in the approximation acquired through Stein's method. It has been noted for the normal approximation that smaller step sizes may yield better bounds, and we obtain the first rigorous results that verify this intuition. For the examples associated to the normal distribution…

泊松-狄利克雷分布对包含过程平稳分布的近似

Poisson-Dirichlet approximation for the stationary distribution of the inclusion process

本文通过改进Stein方法，利用传统导数简化了对包含过程平稳分布的泊松-狄利克雷分布近似误差界分析。

A Note on Load Balancing in Many-Server Heavy-Traffic Regime

In this note, we apply Stein's method to analyze the performance of general load balancing schemes in the many-server heavy-traffic regime. In particular, consider a load balancing system of $N$ servers and the distance of arrival rate to the capacity region is given by $N^{1-α}$ with $α> 1$. We are interested in the performance as $N$ goes to infinity under a large class of policies. We establish different asymptotics under different scalings and conditions. Specifically, (i) If the second moments linearly increase with $N$ with coefficients $σ_a^2$ and $ν_s^2$, then for any $α> 4$, the distr…