Stein-Chen Method Research Articles

Limit laws for self-loops and multiple edges in the configuration model

Nous considerons les boucles et les aretes multiples dans le modele de configuration lorsque la taille du graphe tend vers l’infini. L’interet de ces variables aleatoires est du au fait que le modele de configuration, conditionne a la simplicite, est un graphe aleatoire uniforme avec des degres prescrits. La simplicite correspond a l’absence des boucles et des aretes multiples. Nous montrons que le nombre des boucles et des aretes multiples converge en loi vers deux variables aleatoires independantes qui suivent des lois de Poisson lorsque le moment d’ordre 2 de la loi empirique des degres converge. Nous fournissons aussi des estimations des distances de variation totale entre les nombres des boucles et des aretes multiples et leurs limites, ainsi qu’entre la somme de ces nombres et la variable aleatoire, qui suit une loi de Poisson, vers laquelle converge cette somme. Cela revisite les œuvres precedentes de Bollobas comme de Janson, de Wormald, et d’autres. Les estimations d’erreur impliquent egalement une asymptotique precise pour le nombre de graphes simples avec des degres prescrits. Les estimations d’erreur decoulent d’une application de la methode de Stein–Chen pour la convergence vers une loi de Poisson, qui est une nouvelle methode pour ce probleme. L’independance asymptotique des boucles et des aretes multiples suit a partir d’une version Poisson du dispositif Cramer–Wold utilisant l’amincissement, qui est interessant en lui-meme. Lorsque la loi des degres a un moment d’ordre 2 infini, nos resultats generaux echouent. Nous pouvons, cependant, prouver un theoreme de la limite centrale pour le nombre des boucles, et pour les aretes multiples entre sommets avec degres beaucoup plus petits que la racine carree de la taille du graphe. Nos resultats et preuves peuvent facilement s’etendre aux modeles de configuration orientes et bipartis.

Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates

This article proposes a global, chaos-based procedure for the discretization of functionals of Brownian motion into functionals of a Poisson process with intensity λ>0. Under this discretization we study the weak convergence, as the intensity of the underlying Poisson process goes to infinity, of Poisson functionals and their corresponding Malliavin-type derivatives to their Wiener counterparts. In addition, we derive a convergence rate of O(λ−1∕4) for the Poisson discretization of Wiener functionals by combining the multivariate Chen–Stein method with the Malliavin calculus. Our proposed sufficient condition for establishing the mentioned convergence rate involves the kernel functions in the Wiener chaos, yet we provide examples, especially the discretization of some common path dependent Wiener functionals, to which our results apply without committing the explicit computations of such kernels. To the best our knowledge, these are the first results in the literature on the universal convergence rate of a global discretization of general Wiener functionals.

New Bounds on Poisson Approximation for Random Sums of Independent Binomial Random Variables}

In this paper, we use the Stein-Chen method to obtain new bounds on Poisson approximation for random sums of independent binomial random variables. Some results related to sums of independent binomial distributed random variables are also investigated. The received results in the present study are more general and sharper than some known results.

Multivariate approximation in total variation, I: Equilibrium distributions of Markov jump processes

For integer valued random variables, the translated Poisson distributions form a flexible family for approximation in total variation, in much the same way that the normal family is used for approximation in Kolmogorov distance. Using the Stein–Chen method, approximation can often be achieved with error bounds of the same order as those for the CLT. In this paper, an analogous theory, again based on Stein’s method, is developed in the multivariate context. The approximating family consists of the equilibrium distributions of a collection of Markov jump processes, whose analogues in one dimension are the immigration-death processes with Poisson distributions as equilibria. The method is illustrated by providing total variation error bounds for the approximation of the equilibrium distribution of one Markov jump process by that of another. In a companion paper, it is shown how to use the method for discrete normal approximation in $\mathbb{Z}^{d}$.

Respondent-Driven Sampling and Sparse Graph Convergence

We consider a particular respondent-driven sampling procedure governed by a graphon. Using a specific clumping procedure of the sampled vertices, we construct a sequence of sparse graphs. If the sequence of the vertex-sets is stationary, then the sequence of sparse graphs converges to the governing graphon in the cut-metric. The tools used are a concentration inequality for Markov chains and the Stein-Chen method.

Poisson statistics of eigenvalues in the hierarchical Dyson model

Пусть $(X,d)$ - локально компактное сепарабельное ультраметрическое пространство. Для заданных меры $m$ на $X$ и функции $C$, определенной на множестве $\mathcal{B}$ всех шаров $B\subset X$, рассматривается иерархический лапласиан $L=L_C$. Оператор $L$ действует на пространстве $L^2(X,m)$, является существенно самосопряженным и имеет чисто точечный спектр. Выбор семейства $\{\varepsilon(B)\}_{B\in\mathcal{B}}$ независимых одинаково распределенных случайных величин определяет возмущенную функцию $\mathcal{C}(B)=C(B)(1+\varepsilon(B))$ и возмущенный иерархический лапласиан $\mathcal{L}=L_{\mathcal{C}}$. Все «исходы» возмущенного оператора $\mathcal{L}$ являются иерархическими лапласианами. В частности, все они имеют чисто точечный спектр. Мы изучаем эмпирический точечный процесс $M$, определяемый в терминах собственных значений оператора $\mathcal{L}$. При некоторых естественных предположениях процесс $M$ можно аппроксимировать пуассоновским точечным процессом. Используя результат Р. Арратьи, Л. Гольдштейна и Л. Гордона, основанный на методе Чена-Стейна, мы устанавливаем для пуассоновской аппроксимации скорость сходимости в метрике полной вариации. Нашу теорию мы применяем к случайным возмущениям оператора $\mathfrak{D}^\alpha$, определяемого как $p$-адическая дробная производная порядка $\alpha>0$.

Poisson approximation of the length spectrum of random surfaces

Multivariate Poisson approximation of the length spectrum of random surfaces is studied by means of the Chen-Stein method. This approach delivers simple and explicit error bounds in Poisson limit theorems. They are used to prove that Poisson approximation applies to curves of length up to order $o(\log\log g)$ with $g$ being the genus of the surface.

APPLICATIONS OF STEIN-CHEN METHOD FOR THE PROBLEM OF COINCIDENCES

APPLICATIONS OF STEIN-CHEN METHOD FOR THE PROBLEM OF COINCIDENCES

Symmetric motifs in random geometric graphs

We study symmetric motifs in random geometric graphs. Symmetric motifs are subsets of nodes which have the same adjacencies. These subgraphs are particularly prevalent in random geometric graphs and appear in the Laplacian and adjacency spectrum as sharp, distinct peaks, a feature often found in real-world networks. We look at the probabilities of their appearance and compare these across parameter space and dimension. We then use the Chen-Stein method to derive the minimum separation distance in random geometric graphs which we apply to study symmetric motifs in both the intensive and thermodynamic limits. In the thermodynamic limit the probability that the closest nodes are symmetric approaches one, whilst in the intensive limit this probability depends upon the dimension.

Poisson Approximation of Rademacher Functionals by the Chen-Stein Method and Malliavin Calculus

Poisson Approximation of Rademacher Functionals by the Chen-Stein Method and Malliavin Calculus

Limiting distribution of the number of clumps of palindromes in DNA

ABSTRACTAn important aspect in the analysis of long DNA sequences is to identify whether palindromes are over- or under-represented. The essential step in that direction is the analysis of the limiting distribution of the number of clumps of palindromes where clump is defined as the overlapping occurrence of palindromes. Using the Chen–Stein method, it is shown in this paper that the limiting distribution, under suitable conditions and a type of heterogeneous sequence, is the Poisson distribution. Moreover, error bounds and the rate of convergence are derived in terms of total variation distance between two probability distributions.

Bound on Polya approximation via Stein-Chen's method and w-function

Bound on Polya approximation via Stein-Chen's method and w-function

Extremes of Some Gaussian Random Interfaces

In this article we give a general criterion for some dependent Gaussian models to belong to maximal domain of attraction of Gumbel, following an application of the Stein-Chen method studied in Arratia et al(1989). We also show the convergence of the associated point process. As an application, we show the conditions are satisfied by some of the well-known supercritical Gaussian interface models, namely, membrane model, massive and massless discrete Gaussian free field, fractional Gaussian free field.

Poisson Approximation for a Sum of Negative Binomial Random Variables

The Stein–Chen method is used to determine uniform and non-uniform bounds on the ratio between the distribution function of a sum of independent negative binomial random variables and a Poisson distribution function with mean \(\lambda = \sum _{i=1}^nr_iq_i\), where \(r_i\) and \(p_i=1-q_i\) are parameters of each negative binomial distribution. With these bounds, it indicates that the Poisson distribution function with this mean can be used as an estimate of the independent summands when all \(q_i\) are small or \(\lambda \) is small. Finally, some numerical examples for each result are given.

On bounds in Poisson approximation for distributions of independent negative-binomial distributed random variables.

Using the Stein–Chen method some upper bounds in Poisson approximation for distributions of row-wise triangular arrays of independent negative-binomial distributed random variables are established in this note.

Statistical analysis of the number of self-overlapping leftmost repeats in an homogeneous stationary Markov chain on finite states

ABSTRACTThis article addresses the problem of repeats detection used in the comparison of significant repeats in sequences. The case of self-overlapping leftmost repeats for large sequences generated by an homogeneous stationary Markov chain has not been treated in the literature. In this work, we are interested by the approximation of the number of self-overlapping leftmost long enough repeats distribution in an homogeneous stationary Markov chain. Using the Chen–Stein method, we show that the number of self-overlapping leftmost long enough repeats distribution is approximated by the Poisson distribution. Moreover, we show that this approximation can be extended to the case where the sequences are generated by a m-order Markov chain.

Poisson approximation of subgraph counts in stochastic block models and a graphon model

Small subgraph counts can be used as summary statistics for large random graphs. We use the Stein–Chen method to derive Poisson approximations for the distribution of the number of subgraphs in the stochastic block model which are isomorphic to some fixed graph. We also obtain Poisson approximations for subgraph counts in a graphon-type generalisation of the model in which the edge probabilities are (possibly dependent) random variables supported on a subset of [ 0,1 ]. Our results apply when the fixed graph is a member of the class of strictly balanced graphs.

Extremes of the supercritical Gaussian Free Field

We show that the rescaled maximum of the discrete Gaussian Free Field (DGFF) in dimension larger or equal to 3 is in the maximal domain of attraction of the Gumbel distribution. The result holds both for the infinite-volume field as well as the field with zero boundary conditions. We show that these results follow from an interesting application of the Stein-Chen method from Arratia et al. (1989).

Rates of convergence for extremes of geometric random variables and marked point processes

We use the Stein-Chen method to study the extremal behaviour of univariate and bivariate geometric laws. We obtain a rate for the convergence, to the Gumbel distribution, of the law of the maximum of i.i.d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marshall-Olkin geometric variables as marks and determine bounds on the error of the approximation, in an appropriate probability metric, of the law of the MPPE by that of a Poisson process with same mean measure. We then approximate by another Poisson process with an easier-to-use mean measure and estimate the error of this additional approximation. This work contains and extends results contained in the second author’s PhD thesis (Feidt 2013) under the supervision of Andrew D. Barbour.

A bound on Poisson approximation for independent binomial random variables

This paper uses the Stein-Chen method and the binomial w-functions to derive a non-uniform bound for approximating the cumulative distribution function of a sum of independent binomial random variables, each with parameters mi and pi, by a cumulative Poisson distribution function with mean P n=1 mipi. With this bound, the cumulative Poisson distribution function can be used as an approximation of the cumulative distribution function of the summands when all pi are small.