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We prove an exact fourth moment bound for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result -- that has been elusive for several years -- shows that the so-called `fourth moment phenomenon', first discovered by Nualart and Peccati (2005) in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein's method, Malliavin calculus and Mecke-type formulae, as well as on a methodological breakthrough, consisting in the use of carr\'e-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a non-diffusive framework: as such, it represents a significant extension of the seminal contributions by Ledoux (2012) and Azmoodeh, Campese and Poly (2014). To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of non-linear functionals of a Poisson measure., Comment: 33 pages, final version published as paper no. 4 in Annals of Probability 46 (2018)

This paper establishes new concentration inequalities for random matrices constructed from independent random variables. These results are analogous with the generalized Efron–Stein inequalities developed by Boucheron et al. The proofs rely on the method of exchangeable pairs.

With ${\mathcal{Q}}_{q,n}$ the distribution of $n$ minus the rank of a matrix chosen uniformly from the collection of all $n\times(n+m)$ matrices over the finite field $\mathbb{F}_q$ of size $q\ge2$, and ${\mathcal{Q}}_q$ the distributional limit of ${\mathcal{Q}}_{q,n}$ as $n\rightarrow\infty$, we apply Stein's method to prove the total variation bound $\frac{1}{8q^{n+m+1}}\leq\|{\mathcal{Q}}_{q,n}-{\mathcal{Q}}_q\|_{\mathrm{TV}}\leq\frac{3}{q^{n+m+1}}$. In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices., Comment: Published at http://dx.doi.org/10.1214/13-AOP889 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix. The analysis requires a matrix extension of the scalar concentration theory developed by Sourav Chatterjee using Stein's method of exchangeable pairs. When applied to a sum of independent random matrices, this approach yields matrix generalizations of the classical inequalities due to Hoeffding, Bernstein, Khintchine and Rosenthal. The same technique delivers bounds for sums of dependent random matrices and more general matrix-valued functions of dependent random variables., Published in at http://dx.doi.org/10.1214/13-AOP892 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Stein's method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this paper, we provide some extensions of the theory and three applications: (1) We obtain a concentration inequality for the magnetization in the Curie--Weiss model at critical temperature (where it obeys a nonstandard normalization and super-Gaussian concentration). (2) We derive exact large deviation asymptotics for the number of triangles in the Erd\H{o}s--R\'{e}nyi random graph $G(n,p)$ when $p\ge0.31$. Similar results are derived also for general subgraph counts. (3) We obtain some interesting concentration inequalities for the Ising model on lattices that hold at all temperatures., Comment: Published in at http://dx.doi.org/10.1214/10-AOP542 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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 total variation distance for random variables and the Wasserstein metric for processes as in Barbour and Brown [Stochastic Process. Appl. 43 (1992) 9-31]. Also, through the pseudometric, approximation for certain point processes on a given carrier space is carried out by lifting it to one on a larger space, extending an idea of Arratia, Goldstein and Gordon [Statist. Sci. 5 (1990) 403-434]. The error bound in the general result is similar in form to that for Poisson approximation. As it yields the Stein factor 1/\lambda as in Poisson approximation, it provides good approximation, particularly in cases where \lambda is large. The general result is applied to a number of problems including Poisson process modeling of rare words in a DNA sequence., Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000027

Let $\eta$ be a stationary Harris recurrent Markov chain on a Polish state space $(S, \mathscr{F})$, with stationary distribution $\mu$. Let $\Psi_n:=\sum_{i-1}^n I\{\mu\in S_1\}$ be the number of visits to $S_1\in\mathscr{F}$ by $\eta$, where $S_1$ is “rare” in the sense that $\mu(S_1)$ is “small.” We want to find an approximating compound Poisson distribution for$ \mathscr{L}(\Psi_n)$, such that the approximation error, measured using the total variation distance, can be explicitly bounded with a bound of order not much larger than $\mu(S_1)$. This is motivated by the observation that approximating Poisson distributions often give larger approximation errors when the visits to $S_1$ by $\eta$ tend to occur in clumps and also by the compound Poisson limit theorems of classical extreme value theory. ¶ We here propose an approximating compound Poisson distribution which in a natural way takes into account the regenerative properties of Harris recurrent Markov chains. A total variation distance error bound for this approximation is derived, using the compound Poisson Stein equation of Barbour, Chen and Loh and certain couplings. When the chain has an atom $S_0$ (e.g., a singleton) such that $\mu(S_0)>0$, the bound depends only on much studied quantities like hitting probabilities and expected hitting times, which satisfy Poisson’s equation. As “by-products” we also get upper and lower bounds for the error in the approximation with Poisson or normal distributions. The above results are illustrated by numerical evaluations of the error bound for some Markov chains on finite state spaces.

Let $E$ be a locally compact Hausdorff space with countable basis and let $(X_i)_{\i\in\mathbb{N}}$ be a family of random elements on $E$ with $(1/n) \sum^n_{i=1} \mathscr{L}(X_i) \Rightarrow^v \mu (n \rightarrow \infty)$ for a measure $\mu$ with $\|\mu\| \leq 1$. Conditions are derived under which $\mathscr{L} ((1/n) \sum^n_{i=1} \delta_{Xi}) \Rightarrow^w \delta_\mu(n \rightarrow \infty)$, where $\delta_x$ denotes the Dirac measure at $x$. The proof being based on Stein's method, there are generalisastions that allow for weak dependence between the $X_i$'s. As examples, a dissociated family and an immigration-death process are considered. The latter illustrates the possible applications in proving convergence of stochastic processes.

The aim of this paper is to extend Stein's method to a compound Poisson distribution setting. The compound Poisson distributions of concern here are those of the form POIS$(\nu)$, where $\nu$ is a finite positive measure on $(0, \infty)$. A number of results related to these distributions are established. These in turn are used in a number of examples to give bounds for the error in the compound Poisson approximation to the distribution of a sum of random variables.

Let $X_{n1},\ldots, X_{nn}, n \geq 1$, be independent random variables with $P(X_{ni} = 1) = 1 - P(X_{ni} = 0) = p_{ni}$ such that $\max\{p_{ni}: 1 \leq i \leq n\}\rightarrow 0$ as $n\rightarrow\infty$. Let $W_n = \sum_{1\leq k\leq n} X_{nk}$ and let $Z$ be a Poisson random variable with mean $\lambda = EW_n$. Poisson approximation for the distribution of $W_n$ dates back to 1960, when Le Cam obtained upper bounds for the total variation distance $d(W_n, Z) = \sum_{k\geq 0}|P(W_n = k) - P(Z = k)|$. Barbour and Hall (in 1984) and Deheuvels and Pfeifer (in 1986) investigated the asymptotic behavior of $d(W_n, Z)$ as $n\rightarrow\infty$ for small, moderate and large $\lambda$. Their results imply that the orders of the bounds obtained by Le Cam are best possible. Chen proved (in 1974 and 1975) that the more general variation distance $d(h, W_n, Z) = \sum_{k\geq 0}h(k)|P(W_n = k) - P(Z = k)|, h \geq 0$, converges to 0 as $n\rightarrow\infty$, provided $\lambda$ remains bounded and $Eh(Z) < \infty$. We investigate the asymptotic behavior of: (i) $Eh(W_n) - Eh(Z_\lambda)$ for real $h$; and (ii) $d(h, W_n, Z)$ for $h \geq 0$ as $n\rightarrow\infty$ for small and moderate $\lambda$, thus generalizing the corresponding results of Barbour and Hall and of Deheuvels and Pfeifer. Our method also yields a large deviation result and holds promise for successful application in the case when $X_{n1},\ldots, X_{nn}$ are dependent.

Let $X_1, \cdots, X_n$ be an arbitrary sequence of dependent Bernoulli random variables with $P(X_i = 1) = 1 - P(X_i = 0) = p_i.$ This paper establishes a general method of obtaining and bounding the error in approximating the distribution of $\sum^n_{i=1} X_i$ by the Poisson distribution. A few approximation theorems are proved under the mixing condition of Ibragimov (1959), (1962). One of them yields, as a special case and with some improvement, an approximation theorem of Le Cam (1960) for the Poisson binomial distribution. The possibility of an asymptotic expansion is also discussed and a refinement in the independent case obtained. The method is similar to that of Charles Stein (1970) in his paper on the normal approximation for dependent random variables.

Let $\{X_k\}^\infty_{k = 1}$ be a sequence of independent centered random variables with values in $C(S)$ (i.e., sample continuous processes in $S$), $(S,d)$ being a compact metric space. This sequence is said to satisfy the central limit theorem if there exists a sample continuous Gaussian process on $S, Z$, such that $\mathscr{L}(\sum^n_{k = 1}X_k/n^{\frac{1}{2}})\rightarrow_{w^\ast} \mathscr{L}(Z)$ in $C'(C(S))$. In this paper some sufficient conditions are given for the central limit theorem to hold for $\{X_k\}^\infty_{k = 1}$; these conditions are on the modulus of continuity of the processes $X_k$ and they are expressed in terms of the metric entropy of distances associated to $\{X_k\}$. Then, in order to give some insight on these theorems, several results on the central limit theorem for particular processes (random Fourier and Taylor series, as well as more general processes on [0, 1]) are deduced.