We obtain explicit error bounds for the $d$-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a non-linear statistic of independent random variables or a sum of $n$ locally dependent random vectors. We assume the approximating normal distribution has a non-singular covariance matrix. The error bounds vanish even when the dimension $d$ is much larger than the sample size $n$. We prove our main results using the approach of Götze (1991) in Stein's method, together with modifications of an estimate of Anderson, Hall and Titterington (1998) and a smoothing inequality of Bhattacharya and Rao (1976). For sums of $n$ independent and identically distributed isotropic random vectors having a log-concave density, we obtain an error bound that is optimal up to a $\log n$ factor. We also discuss an application to multiple Wiener-Itô integrals.
In this article we propose a general framework for normal approximation using Stein's method. We introduce the new concept of Stein couplings and we show that it lies at the heart of popular approaches such as the local approach, exchangeable pairs, size biasing and many other approaches. We prove several theorems with which normal approximation for the Wasserstein and Kolmogorov metrics becomes routine once a Stein coupling is found. To illustrate the versatility of our framework we give applications in Hoeffding's combinatorial central limit theorem, functionals in the classic occupancy sche…
We provide an explanation of the main ideas underlying Götze's main result in using Stein's method. We also provide detailed derivations of various intermediate estimates. Curiously, we are led to a different dimensional dependence of the constant than that given Götze's paper. We would like to dedicate this to Charles Stein on the occasion of his 90th birthday.
Consider the multivariate Stein equation $Δf - x\cdot \nabla f = h(x) - E h(Z)$, where $Z$ is a standard $d$-dimensional Gaussian random vector, and let $f\_h$ be the solution given by Barbour's generator approach. We prove that, when $h$ is $α$-Hölder ($0<α\leq1$), all derivatives of order $2$ of $f\_h$ are $α$-Hölder {\it up to a $\log$ factor}; in particular they are $β$-Hölder for all $β\in (0, α)$, hereby improving existing regularity results on the solution of the multivariate Gaussian Stein equation. For $α=1$, the regularity we obtain is optimal, as shown by an example given by Raič \c…
The fourth moment theorem provides error bounds of the order $\sqrt{{\mathbb E}(F^4) - 3}$ in the central limit theorem for elements $F$ of Wiener chaos of any order such that ${\mathbb E}(F^2) = 1$. It was proved by Nourdin and Peccati (2009) using Stein's method and the Malliavin calculus. It was also proved by Azmoodeh, Campese and Poly (2014) using Stein's method and Dirichlet forms. This paper is an exposition on the connections between Stein's method and the Malliavin calculus and between Stein's method and Dirichlet forms, and on how these connections are exploited in proving the fourth…
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 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 th…
We generalize the well-known zero bias distribution and the $λ$-Stein pair to an approximate zero bias distribution and an approximate $λ,R$-Stein pair, respectively. Berry Esseen type bounds to the normal, based on approximate zero bias couplings and approximate $λ,R$-Stein pairs, are obtained using Stein's method. The bounds are then applied to combinatorial central limit theorems where the random permutation has the Ewens $\mathcal{E}_θ$ distribution with $θ>0$ which can be specialized to the uniform distribution by letting $θ=1$. The family of the Ewens distributions appears in the context…
We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in the context of the Hoeffding combinatorial central limit theorem with integer valued summands, of the number of isolated vertices in an Erdős-Rényi random graph, and of the Curie-Weiss model of magnetism, where we provide optimal or near optimal rates of convergence in the local limit metric. In the Hoeffding example, even the discrete normal approximation …
We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution {which is based on a linear difference or differential-type operator}. The resulting Stein identity highlights the unifying theme behind the literature on Stein's method (both for continuous and discrete distributions). Viewing the Stein operator as an operator acting on pairs of functions, we provide an extensive toolkit for distributional comparisons. Several abstract approximation theorems are provided. Our appro…
We develop Stein's method for the half-normal distribution and apply it to derive rates of convergence in distributional limit theorems for three statistics of the simple symmetric random walk: the maximum value, the number of returns to the origin and the number of sign changes up to a given time $n$. We obtain explicit error bounds with the optimal rate $n^{-1/2}$ for both the Kolmogorov and the Wasserstein metric. In order to apply Stein's method, we compare the characterizing operator of the limiting half-normal distribution with suitable characterizations of the discrete approximating dis…
We use Stein's method to bound the Wasserstein distance of order $2$ between a measure $ν$ and the Gaussian measure using a stochastic process $(X_t)_{t \geq 0}$ such that $X_t$ is drawn from $ν$ for any $t > 0$. If the stochastic process $(X_t)_{t \geq 0}$ satisfies an additional exchangeability assumption, we show it can also be used to obtain bounds on Wasserstein distances of any order $p \geq 1$. Using our results, we provide optimal convergence rates for the multi-dimensional Central Limit Theorem in terms of Wasserstein distances of any order $p \geq 2$ under simple moment assumptions.
In the seventies, Charles Stein revolutionized the way of proving the Central Limit Theorem by introducing a method that utilizes a characterization equation for Gaussian distribution. In the last 50 years, much research has been done to adapt and strengthen this method to a variety of different settings and other limiting distributions. However, it has not been yet extended to study conditional convergences. In this article, we develop a novel approach using Stein's method for exchangeable pairs to find a rate of convergence in Conditional Central Limit Theorem of the form $(X_n\mid Y_n=k)$, …
Suppose that $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}$ is the solution to a $d$-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalang's condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form $N^{-d} \int_{[0,N]^d} g(u(t\,,x))\, \mathrm{d} x$, as $N\rightarrow\infty$, where $g$ is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Stein's method for …
We consider the imitative monomer-dimer model on the complete graph introduced in [1]. It was understood that this model is described by the monomer density and has a phase transition along certain critical line. By reverting the model to a weighted Curie-Weiss model with hard core interaction, we establish the complete description of the fluctuation properties of the monomer density on the full parameter space via Stein's method of exchangeable pairs. We show that this quantity exhibits the central limit theorem away from the critical line and enjoys a non-normal limit theorem at criticality …
Motivated by a theorem of Barbour, we revisit some of the classical limit theorems in probability from the viewpoint of the Stein method. We setup the framework to bound Wasserstein distances between some distributions on infinite dimensional spaces. We show that the convergence rate for the Poisson approximation of the Brownian motion is as expected proportional to $λ^{-1/2}$ where $λ$ is the intensity of the Poisson process. We also exhibit the speed of convergence for the Donsker Theorem and for the linear interpolation of the Brownian motion. By iterating the procedure, we give Edgeworth e…
We study the Stein equation associated with the one-dimensional Gamma distribution, and provide novel bounds, allowing one to effectively deal with test functions supported by the whole real line. We apply our estimates to derive new quantitative results involving random variables that are non-linear functionals of random fields, namely: (i) a non-central quantitative de Jong theorem for sequences of degenerate U-statistics satisfying minimal uniform integrability conditions, significantly extending previous findings by de Jong (1990), Nourdin, Peccati and Reinert (2010) and Döbler and Peccati…
Given an isotropic probability measure $μ$ on ${\mathbb R}^d$ with ${\rm d}μ\left( x \right) = {\left( {\varrho \left( x \right)} \right)^{ - α}}{\rm d}x$, where $α> d + 1$ and $\varrho :{{\mathbb R}^d} \to \left( {0, + \infty } \right)$ is a continuous function and uniformly convex (${\nabla ^2}\varrho \ge {\varepsilon_0}{\rm {Id}}$). By using Stein kernels for $\left( {α- d} \right)$-moment measures, we prove that the rates of convergence in the central limit theorem with sequence of i.i.d. random variables ${X_1},{X_2},...,{X_n}$ of the law $μ$, to be of form $c_{{\varepsilon}_0}\,\sqrt {\d…
Stein's method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second- or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismut's formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein's exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wass…
We obtain almost optimal convergence rate in the central limit theorem for "nonconevntional" sums of the form $S_N=N^{-\frac12}\sum_{n=1}^N (F(ξ_n,ξ_{2n},...,ξ_{\ell n})-\bar F)$.
We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for the number of germs with $d$ neighbours in a germ--grain model, and the number of degree-$d$ vertices in an Erdős--Rényi random graph. In both cases, the error rate is optimal, up to logarithmic f…
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 paper, Cramér-type moderate deviation theorems are established for nonnormal approximation under a general Stein identity, which is satisfied via the exchangeable pair approach and Stein's coupling. In particular…
Stein's method allows to prove distributional convergence of a sequence of random variables and to quantify it with respect to a given metric such as Kolmogorov's (a Berry-Esséen type theorem). Mod-* convergence quantifies the convergence of a sequence of random variables to a given distribution in a sense unusual in probability theory, a priori unrelated to a metric on probability measures. This article gives a connection between these two notions. It shows that mod-* convergence can be understood as a higher order approximation in distribution when the limiting function is integrable and pro…
Stein's method is a method of probability approximation which hinges on the solution of a functional equation. For normal approximation the functional equation is a first order differential equation. Malliavin calculus is an infinite-dimensional differential calculus whose operators act on functionals of general Gaussian processes. Nourdin and Peccati (2009) established a fundamental connection between Stein's method for normal approximation and Malliavin calculus through integration by parts. This connection is exploited to obtain error bounds in total variation in central limit theorems for …
The original Donsker theorem says that a standard random walk converges in distribution to a Brownian motion in the space of continuous functions. It has recently been extended to enriched random walks and enriched Brownian motion. We use the Stein-Dirichlet method to precise the rate of this convergence in the topology of fractional Sobolev spaces.
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 theorem augmented by a rate of convergence. We then present a scheme for checking these conditions in actual examples. The principal contribution of our paper is the method, which yields a convergence rate essentially with the same amount of work as the central limit theorem, together with a multiplicative constant that can be computed directly from the assumption…
We combine Stein's method with a version of Malliavin calculus on the Poisson space. As a result, we obtain explicit Berry-Esséen bounds in Central Limit Theorems (CLTs) involving multiple Wiener-Itô integrals with respect to a general Poisson measure. We provide several applications to CLTs related to Ornstein-Uhlenbeck Lévy processes.
Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any nonnegative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given deg…
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 …
This paper is a short exposition of Stein's method of normal approximation from my personal perspective. It focuses mainly on the characterization of the normal distribution and the construction of Stein identities. Through examples, it provides glimpses into the many approaches to constructing Stein identities and the diverse applications of Stein's method to mathematical problems. It also includes anecdotes of historical interest, including how Stein discovered his method and how I found an unpublished proof of his of the Berry-Esseen theorem.
We combine the method of exchangeable pairs with Stein's method for functional approximation. As a result, we give a general linearity condition under which an abstract Gaussian approximation theorem for stochastic processes holds. We apply this approach to estimate the distance of a sum of random variables, chosen from an array according to a random permutation, from a Gaussian mixture process. This result lets us prove a functional combinatorial central limit theorem. We also consider a graph-valued process and bound the speed of convergence of the distribution of its rescaled edge counts to…
We prove a general theorem to bound the total variation distance between the distribution of an integer valued random variable of interest and an appropriate discretized normal distribution. We apply the theorem to 2-runs in a sequence of i.i.d. Bernoulli random variables, the number of vertices with a given degree in the Erdös-Rényi random graph, and the uniform multinomial occupancy model.
We derive a new result for exponential approximation using Stein's method of exchangeable pairs. As an application, an exponential limit theorem with error term is derived for |Tr(U)|^2, where Tr(U) denotes the trace of a matrix chosen from the Haar measure of the unitary group U(n,C).
The one parameter family of Jack(alpha) measures on partitions is an important discrete analog of Dyson's beta ensembles of random matrix theory. Except for special values of alpha=1/2,1,2 which have group theoretic interpretations, the Jack(alpha) measure has been difficult if not intractable to analyze. This paper proves a central limit theorem (with an error term) for Jack(alpha) measure which works for arbitrary values of alpha. For alpha=1 we recover a known central limit theorem on the distribution of character ratios of random representations of the symmetric group on transpositions. Th…
We combine Malliavin calculus with Stein's method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. We also prove results concerning random variables admitting a possibly infinite Wiener chaotic decomposition. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener-Itô integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry-Esséen bounds in…
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 equation in the case $n=1$. This Stein equation motivates a generalisation of the zero bias transformation. We establish properties of this new transformation, and illustrate how they may be used together with the Stein equation to assess distributional distances for statistics that are asymptotically distributed as the product of independent central normal random…
Using Stein's method techniques, we develop a framework which allows one to bound the error terms arising from approximation by the Laplace distribution and apply it to the study of random sums of mean zero random variables. As a corollary, we deduce a Berry-Esseen type theorem for the convergence of certain geometric sums. Our results make use of a second order characterizing equation and a distributional transformation which is related to zero-biasing.
In his work \cite{Ti80}, Tikhomirov combined elements of Stein's method with the theory of characteristic functions to derive Kolmogorov bounds for the convergence rate in the central limit theorem for a normalized sum of a stationary sequence of random variables satisfying one of several weak dependency conditions. The combination of elements of Stein's method with the theory of characteristic functions is sometimes called \emph{Stein-Tikhomirov method}. \citet*{AMPS17} successfully used the Stein-Tikhomirov method to bound the convergence rate in contexts with non-Gaussian targets. \citet*{R…
In this paper, we extend Stein's method to products of independent beta, gamma, generalised gamma and mean zero normal random variables. In particular, we obtain Stein operators for mixed products of these distributions, which include the classical beta, gamma and normal Stein operators as special cases. These operators lead us to closed-form expressions involving the Meijer $G$-function for the probability density function and characteristic function of the mixed product of independent beta, gamma and central normal random variables.
We introduce some applications of Stein's method in the high temperature analysis of spin glasses. Stein's method allows the direct analysis of the Gibbs measure without having to create a cavity. Another advantage is that it gives limit theorems with total variation error bounds, although the bounds can be suboptimal. A surprising byproduct of our analysis is a relatively transparent explanation of the Thouless-Anderson-Palmer system of equations. Along the way, we develop Stein's method for mixtures of two Gaussian densities.
