Stein's (1972) method is a very general tool for assessing the quality of approximation of the distribution of a random element by another, often simpler, distribution. In applications of Stein's method, one needs to establish a Stein identity for the approximating distribution, solve the Stein equation and estimate the behaviour of the solutions in terms of the metrics under study. For some Stein equations, solutions with good properties are known; for others, this is not the case. Barbour and Xia (1999) introduced a perturbation method for Poisson approximation, in which Stein identities for a large class of compound Poisson and translated Poisson distributions are viewed as perturbations of a Poisson distribution. In this paper, it is shown that the method can be extended to very general settings, including perturbations of normal, Poisson, compound Poisson, binomial and Poisson process approximations in terms of various metrics such as the Kolmogorov, Wasserstein and total variation metrics. Examples are provided to illustrate how the general perturbation method can be applied.
In this paper, we give an upper bound for a probabilistic distance between a Gaussian vector and a vector of U-statistics of Poisson point processes by applying Malliavin-Stein inequality on the Poisson space.
We obtain Stein approximation bounds for stochastic integrals with respect to a Poisson random measure over ${\Bbb R}^d$, $d\geq 2$. This approach relies on third cumulant Edgeworth-type expansions based on derivation operators defined by the Malliavin calculus for Poisson random measures. The use of third cumulants can exhibit faster convergence rates than the standard Berry-Esseen rate for some sequences of Poisson stochastic integrals.
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.
We establish various bounds on the solutions to a Stein equation for Poisson approximation in Wasserstein distance with non-linear transportation costs. The proofs are a refinement of those in [Barbour and Xia (2006)] using the results in [Liu and Ma (2009)]. As a corollary, we obtain an estimate of Poisson approximation error measured in L^2-Wasserstein distance.
One major obstacle in applications of Stein's method for compound Poisson approximation is the availability of so-called magic factors (bounds on the solution of the Stein equation) with favourable dependence on the parameters of the approximating compound Poisson random variable. In general, the best such bounds have an exponential dependence on these parameters, though in certain situations better bounds are available. In this paper, we extend the region for which well-behaved magic factors are available for compound Poisson approximation in the Kolmogorov metric, allowing useful compound Po…
In this paper, we apply the Stein's method in the context of point processes, namely when the target measure is the distribution of a finite Poisson point process. We show that the so-called Kantorovich-Rubinstein distance between such a measure and another finite point process is bounded by the $L^1$-distance between their respective Papangelou intensities. Then, we deduce some convergence rates for sequences of point processes approaching a Poisson or a Cox point process.
The main purpose of the paper is to investigate the possibility of applying Chen-Stein approach to estimate the $χ^2$ distance between Poisson distribution and a sum of independent indicators. Earlier results concerning $χ^2$ distance between above mentioned distributions either used analytical approach heavily based on the analysis of the generating functions or on rather lengthy and complicated elementary calculations. Applying Chen-Stein approach we succeed in providing a very quick proof of upper bounds for $χ^2$ distance that are of comparable strength to the earlier estimates obtained by…
This exposition explains the basic ideas of Stein's method for Poisson random variable approximation and Poisson process approximation from the point of view of the immigration-death process and Palm theory. The latter approach also enables us to define local dependence of point processes [Chen and Xia (2004)] and use it to study Poisson process approximation for locally dependent point processes and for dependent superposition of point processes.
We establish a general inequality on the Poisson space, yielding an upper bound for the distance in total variation between the law of a regular random variable with values in the integers and a Poisson distribution. Several applications are provided, in particular: (i) to deduce a set of sufficient conditions implying that a sequence of (suitably shifted) multiple Wiener-Itô integrals converges in distribution to a Poisson random variable, and (ii) to compute explicit rates of convergence for the Poisson approximation of statistics associated with geometric random graphs with sparse connectio…
This paper uses the generator comparison approach of Stein's method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. The "standard" generator comparison approach starts with the Poisson equation for the diffusion, and the main technical difficulty is to obtain bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the Poisson equation of the Markov chain; we term this the prelimit approach. Although one still needs Stein factor bounds, they now correspond to fini…
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 combination of the Chen-Stein method and a discrete version of Malliavin calculus. We give some applications to shifted discrete multiple stochastic integrals.
In this article, we provide an extension of the Chen-Stein inequality for Poisson approximation in the total variation distance for sums of independent Bernoulli random variables in two ways. We prove that we can improve the rate of convergence (hence the quality of the approximation) by using explicitly constructed signed or positive probability measures, and that we can extend the setting to possibly dependent random variables. The framework which allows this is that of mod-Poisson convergence, and more precisely those mod-Poisson convergent sequences whose residue functions can be expressed…
We apply the Stein-Chen method to problems from extreme value theory. On the one hand, the Stein-Chen method for Poisson approximation allows us to obtain bounds on the Kolmogorov distance between the law of the maximum of i.i.d. random variables, following certain well known distributions, and an extreme value distribution. On the other hand, we introduce marked point processes of exceedances (MPPE's) whose i.i.d. marks can be either univariate or multivariate. We use the Stein-Chen method for Poisson process approximation to determine bounds on the error of the approximation, in some appropr…
The framework of Stein's method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem \refimportantproposition) in Poisson process approximation is proved by taking the local approach. It is obtained without reference to any particular metric, thereby allowing wider applicability. A Wasserstein pseudometric is introduced for measuring the accuracy of point process approximation. The pseudometric provides a generalization of many metrics used so far, including the t…
Based on Stein's method, we derive upper bounds for Poisson process approximation in the $L_1$-Wasserstein metric $d_2^{(p)}$, which is based on a slightly adapted $L_p$-Wasserstein metric between point measures. For the case $p=1$, this construction yields the metric $d_2$ introduced in [Barbour and Brown Stochastic Process. Appl. 43 (1992) 9--31], for which Poisson process approximation is well studied in the literature. We demonstrate the usefulness of the extension to general $p$ by showing that $d_2^{(p)}$-bounds control differences between expectations of certain $p$th order average stat…
Point processes are an essential tool when we are interested in where in time or space events occur. The basic starting point for point processes is usually the Poisson process. Over the years, Stein's method has been developed with a great deal of success for Poisson point process approximation. When studying rare events though, typically one only begins modelling after the occurrence of such an event. As a result, a point process that is conditional upon at least one atom, is arguably more appropriate in certain applications. In this paper, we develop Stein's method for conditional Poisson p…
In this paper, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions. Also, Stein operators for certain compound distributions, where the random summand satisfies Panjer's recurrence relation, are derived. A well-known perturbation approach for Stein's method is used to obtain total variation bounds for the distributions mentioned above. The importance of such approximations is illustrated, for example, by the binomial convoluted with Poisson approxi…
Random events in space and time often exhibit a locally dependent structure. When the events are very rare and dependent structure is not too complicated, various studies in the literature have shown that Poisson and compound Poisson processes can provide adequate approximations. However, the accuracy of approximations does not improve or may even deteriorate when the mean number of events increases. In this paper, we investigate an alternative family of approximating point processes and establish Stein's method for their approximations. We prove two theorems to accommodate respectively the po…
A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on Poisson and translated Poisson approximation of a sum of dependent Bernoulli random variables, for example k-runs in i.i.d. Bernoulli trials. Other applications include approximation by polynomial birth--death distributions.
This paper uses the generator approach of Stein's method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. Until now, the standard way to invoke Stein's method for this problem was to use the Poisson equation for the diffusion as a starting point. The main technical difficulty with this approach is obtaining bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the discrete Poisson equation of the Markov chain. An important step in our approach is extending the d…
In this paper we use a Malliavin-Stein type method to investigate Poisson and normal approximations for the measurable functions of infinitely many independent random variables. We combine Stein's method with the difference operators in theory of concentration inequalities to obtain explicit bounds on Wasserstein, Kolmogorov and total variation distances. When restricted to the functions of a finite number of independent random variables, our method provides new bounds in the normal approximation. Meanwhile, our bounds in Poisson approximation are first to obtain explicitly.
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 provide a general theorem bounding the error in the approximation of a random measure of interest--for example, the empirical population measure of types in a Wright-Fisher model--and a Dirichlet process, which is a measure having Poisson-Dirichlet distributed atoms with i.i.d. labels from a diffuse distribution. The implicit metric of the approximation theorem captures the sizes and locations of the masses, and so also yields bounds on the approximation between the masses of the measure of interest and the Poisson-Dirichlet distribution. We apply the result to bound the error in the approx…
We study an approximation method of stationary characters of a two-dimensional Markov chain via the Stein method. For this purpose, innovative methods are developed to estimate the moments of the Markov chain, as well as the solution to the Poisson equation with a partial differential operator.
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 …
In this paper, we obtain Stein operator for sum of $n$ independent random variables (rvs) which is shown as perturbation of negative binomial (NB) operator. Comparing the operator with NB operator, we derive the error bounds for total variation distance by matching parameters. Also, three parameters approximation for such a sum is considered and is shown to improve the existing bounds in the literature. Finally, an application of our results to a function of waiting time for $(k_1,k_2)$-events is given.
Stein's method provides a way of bounding the distance of a probability distribution to a target distribution $μ$. Here we develop Stein's method for the class of discrete Gibbs measures with a density $e^V$, where $V$ is the energy function. Using size bias couplings, we treat an example of Gibbs convergence for strongly correlated random variables due to Chayes and Klein [Helv. Phys. Acta 67 (1994) 30--42]. We obtain estimates of the approximation to a grand-canonical Gibbs ensemble. As side results, we slightly improve on the Barbour, Holst and Janson [Poisson Approximation (1992)] bounds f…
We extend the ideas of (Barbour 1990) and use Stein's method to obtain a bound on the distance between a scaled time-changed random walk and a time-changed Brownian Motion. We then apply this result to bound the distance between a time-changed compensated scaled Poisson process and a time-changed Brownian Motion. This allows us to bound the distance between a process whose dynamics resemble those of the Moran model with mutation and a process whose dynamics resemble those of the Wright-Fisher diffusion with mutation upon noting that the former may be expressed as a difference of two time-chang…
Using the Chen-Stein method, we show that the spatial distribution of large finite clusters in the supercritical FK model approximates a Poisson process when the ratio weak mixing property holds.
This article deals with Stein characterizations of probability distributions. We provide a general framework for interpreting these in terms of the parameters of the underlying distribution. In order to do so we introduce two concepts (a class of functions and an operator) which generalize those which were developed in the 70's by Charles Stein and Louis Chen for characterizing the Gaussian and the Poisson distributions. Our methodology (i) allows for writing many (if not all) known univariate Stein characterizations, (ii) permits to identify clearly minimal conditions under which these result…
As an application of Stein's method for Poisson approximation, we prove rates of convergence for the tail probabilities of two scan statistics that have been suggested for detecting local signals in sequences of independent random variables subject to possible change-points. Our formulation deals simultaneously with ordinary and with large deviations.
We show by a surprisingly simple argument that the exchangeability condition, which is key to the exchangeable pair approach in Stein's method for distributional approximation, can be omitted in many standard settings. This is achieved by replacing the usual antisymmetric function by a simpler one, for which only equality in distribution is required. In the case of normal approximation we also slightly improve the constants appearing in previous results. For Poisson approximation, a different antisymmetric function is used, and additional error terms are needed if the bound is to be extended b…
In this article, we obtain, for the total variance distance, the error bounds between Poisson and convolution of power series distributions via Stein's method. This provides a unified approach to many known discrete distributions. Several Poisson limit theorems follow as corollaries from our bounds. As applications, we compare the Poisson approximation results with the negative binomial approximation results, for the sums of Bernoulli, geometric, and logarithmic series random variables.
We study multi-dimensional normal approximations on the Poisson space by means of Malliavin calculus, Stein's method and probabilistic interpolations. Our results yield new multi-dimensional central limit theorems for multiple integrals with respect to Poisson measures -- thus significantly extending previous works by Peccati, Solé, Taqqu and Utzet. Several explicit examples (including in particular vectors of linear and non-linear functionals of Ornstein-Uhlenbeck Lévy processes) are discussed in detail.
We prove, using the Brascamp-Lieb inequality, that the Gaussian measure is the only strong log-concave measure having a strong log-concavity parameter equal to its covariance matrix. We also give a similar characterization of the Poisson measure in the discrete case, using "Chebyshev's other inequality". We briefly discuss how these results relate to Stein and Stein-Chen methods for Gaussian and Poisson approximation, and to the Bakry-Emery calculus.
Poisson approximation using Stein's method has been extensively studied in the literature. The main focus has been on bounding the total variation distance. This paper is a first attempt on moderate deviations in Poisson approximation for right-tail probabilities of sums of dependent indicators. We obtain results under certain general conditions for local dependence as well as for size-bias coupling. These results are then applied to independent indicators, 2-runs, and the matching problem.
This paper considers the entropy of the sum of (possibly dependent and non-identically distributed) Bernoulli random variables. Upper bounds on the error that follows from an approximation of this entropy by the entropy of a Poisson random variable with the same mean are derived. The derivation of these bounds combines elements of information theory with the Chen-Stein method for Poisson approximation. The resulting bounds are easy to compute, and their applicability is exemplified. This conference paper presents in part the first half of the paper entitled "An information-theoretic perspectiv…
The first part of this work considers the entropy of the sum of (possibly dependent and non-identically distributed) Bernoulli random variables. Upper bounds on the error that follows from an approximation of this entropy by the entropy of a Poisson random variable with the same mean are derived via the Chen-Stein method. The second part of this work derives new lower bounds on the total variation (TV) distance and relative entropy between the distribution of the sum of independent Bernoulli random variables and the Poisson distribution. The starting point of the derivation of the new bounds i…
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…
This paper provides an introduction to the Stein method framework in the context of steady-state diffusion approximations. The framework consists of three components: the Poisson equation and gradient bounds, generator coupling, and moment bounds. Working in the setting of the Erlang-A and Erlang-C models, we prove that both Wasserstein and Kolmogorov distances between the stationary distribution of a normalized customer count process, and that of an appropriately defined diffusion process decrease at a rate of $1/\sqrt{R}$, where $R$ is the offered load. Futhermore, these error bounds are \em…
Consider a graph on randomly scattered points in an arbitrary space, with two points $x,y$ connected with probability $ϕ(x,y)$. Suppose the number of points is large but the mean number of isolated points is $O(1)$. We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Poisson appro…
Donsker Theorem is perhaps the most famous invariance principle result for Markov processes. It states that when properly normalized, a random walk behaves asymptotically like a Brownian motion. This approach can be extended to general Markov processes whose driving parameters are taken to a limit, which can lead to insightful results in contexts like large distributed systems or queueing networks. The purpose of this paper is to assess the rate of convergence in these so-called diffusion approximations, in a queueing context. To this end, we extend the functional Stein method introduced for t…
Using Stein's method and the Malliavin calculus of variations, we derive explicit estimates for the Gamma approximation of functionals of a Poisson measure. In particular, conditions are presented under which the distribution of a sequence of multiple Wiener-Ito stochastic integrals with respect to a compensated Poisson measure converges to a Gamma distribution. As an illustration, we present a quantitative version and a non-central extension of a classical theorem by de Jong in the case of degenerate U-statistics of order two. Several multidimensional extensions, in particular allowing for mi…
We consider $M/Ph/n+M$ queueing systems in steady state. We prove that the Wasserstein distance between the stationary distribution of the normalized system size process and that of a piecewise Ornstein-Uhlenbeck (OU) process is bounded by $C/\sqrtλ$, where the constant $C$ is independent of the arrival rate $λ$ and the number of servers $n$ as long as they are in the Halfin-Whitt parameter regime. For each integer $m>0$, we also establish a similar bound for the difference of the $m$th steady-state moments. For the proofs, we develop a modular framework that is based on Stein's method. The fr…
By a method inspired of the Stein's method, we derive an upper-bound of the Rubinstein distance between two absolutely continuous probability measures on configurations space. As an application, we show that the best way to approximate a Modulated Poisson Process (see below for the definition) by a Poisson process is to equate their intensity.
The paper is concerned with the equilibrium distributions of continuous-time density dependent Markov processes on the integers. These distributions are known typically to be approximately normal, and the approximation error, as measured in Kolmogorov distance, is of the smallest order that is compatible with their having integer support. Here, an approximation in the much stronger total variation norm is established, without any loss in the asymptotic order of accuracy; the approximating distribution is a translated Poisson distribution having the same variance and (almost) the same mean. Our…
This is a survey paper on Poisson approximation using Stein's method of exchangeable pairs. We illustrate using Poisson-binomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector's problem, and the birthday problem. While many details are new, the results are closely related to a body of work developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry Goldstein, and their collaborators. Some comparison with these other approaches is offered.
We compute explicit upper bounds on the distance between the law of a multivariate Gaussian distribution and the joint law of wavelets/needlets coefficients based on a homogeneous spherical Poisson field. In particular, we develop some results from Peccati and Zheng (2011), based on Malliavin calculus and Stein's methods, to assess the rate of convergence to Gaussianity for a triangular array of needlet coefficients with growing dimensions. Our results are motivated by astrophysical and cosmological applications, in particular related to the search for point sources in Cosmic Rays data.
