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

We derive a new compound Poisson distribution with explicit parameters to approximate the number of overlapping occurrences of any set of words in a Markovian sequence. Using the Chen-Stein method, we provide a bound for the approximation error. This error converges to 0 under the rare event condition, even for overlapping families, which improves previous results. As a consequence, we also propose Poisson approximations for the declumped count and the number of competing renewals.

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A Phase Transition for the Distribution of Matching Blocks

Combinatorics Probability Computing ◽

10.1017/s0963548300001930 ◽

1996 ◽

Vol 5 (2) ◽

pp. 139-159 ◽

Cited By ~ 2

Author(s):

Claudia Neuhauser

Keyword(s):

Phase Transition ◽

Law Of Large Numbers ◽

Poisson Approximation ◽

Finite Alphabet ◽

Strong Law ◽

Large Numbers ◽

Stein Method

We show distributional results for the length of the longest matching consecutive subsequence between two independent sequences A1, A2, …, Am and B1, B2, …, Bn whose letters are taken from a finite alphabet. We assume that A1, A2, … are i.i.d. with distribution μ and B1, B2, … are i.i.d. with distribution ν. It is known that if μ and v are not too different, the Chen–Stein method for Poisson approximation can be used to establish distributional results. We extend these results beyond the region where the Chen–Stein method was previously successful. We use a combination of ‘matching by patterns’ results obtained by Arratia and Waterman [1], and the Chen–Stein method to show that the Poisson approximation can be extended. Our method explains how the matching is achieved. This provides an explanation for the formulas in Arratia and Waterman [1] and thus answers one of the questions posed in comment F19 in Aldous [2]. Furthermore, in the case where the alphabet consists of only two letters, the phase transition observed by Arratia and Waterman [1] for the strong law of large numbers extends to the distributional result. We conjecture that this phase transition on the distributional level holds for any finite alphabet.

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Poisson Approximation and the Chen-Stein Method

Statistical Science ◽

10.1214/ss/1177012015 ◽

1990 ◽

Vol 5 (4) ◽

pp. 403-424 ◽

Cited By ~ 185

Author(s):

Richard Arratia ◽

Larry Goldstein ◽

Louis Gordon

Keyword(s):

Poisson Approximation ◽

Stein Method

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On asymptotic properties of the distribution of the number of pairs of H-connected tuples

Discrete Mathematics and Applications ◽

10.1515/dma-2002-0407 ◽

2002 ◽

Vol 12 (4) ◽

Author(s):

V. G. Mikhailov

Keyword(s):

Asymptotic Properties ◽

Sufficient Conditions ◽

Basic Research ◽

Poisson Approximation ◽

Explicit Estimate ◽

Research Grants ◽

Local Variant ◽

Sufficient Conditions For Convergence ◽

Stein Method ◽

Independent Identically Distributed

AbstractThe main result of this paper is a theorem about convergence of the distribution of the number of pairs of H-connected s-tuples in two independent sequences of independent identically distributed variables. The concept of H-connection is a generalisation of the concept of H-equivalence of tuples. We give sufficient conditions for convergence and an explicit estimate of the rate of convergence. We use the local variant of the Chen-Stein method for estimating the accuracy of Poisson approximation for distribution of the set of dependent random indicators. The main results of this paper were announced in [7].The research was supported by the Russian Foundation for Basic Research, grants 02-01-00266 and 00-15-96136.

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A Poisson Approximation for an Occupancy Problem with Collisions

Journal of Applied Probability ◽

10.1017/s0021900200004332 ◽

2008 ◽

Vol 45 (02) ◽

pp. 430-439 ◽

Cited By ~ 1

Author(s):

Toshio Nakata

Keyword(s):

Factorial Moment ◽

Poisson Approximation ◽

Occupancy Problem ◽

Stein Method

We study collision probabilities concerning the simple balls-and-bins problem developed by Wendl (2003). In this article we give the factorial moment of the number of collisions. Moreover, we obtain a Poisson approximation for the number of collisions using the Chen-Stein method.

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Poisson Approximation for the Number of Repeats in a Stationary Markov Chain

Journal of Applied Probability ◽

10.1017/s0021900200004344 ◽

2008 ◽

Vol 45 (02) ◽

pp. 440-455

Author(s):

Narjiss Touyar ◽

Sophie Schbath ◽

Dominique Cellier ◽

Hélène Dauchel

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Poisson Distribution ◽

Statistical Methods ◽

Evolutionary History ◽

Approximation Error ◽

Poisson Approximation ◽

Complete Genomes ◽

Genome Dynamics ◽

Stein Method

Detection of repeated sequences within complete genomes is a powerful tool to help understanding genome dynamics and species evolutionary history. To distinguish significant repeats from those that can be obtained just by chance, statistical methods have to be developed. In this paper we show that the distribution of the number of long repeats in long sequences generated by stationary Markov chains can be approximated by a Poisson distribution with explicit parameter. Thanks to the Chen-Stein method we provide a bound for the approximation error; this bound converges to 0 as soon as the length n of the sequence tends to ∞ and the length t of the repeats satisfies n 2ρ t = O(1) for some 0 < ρ < 1. Using this Poisson approximation, p-values can then be easily calculated to determine if a given genome is significantly enriched in repeats of length t.

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Local matching of random restriction maps

Journal of Applied Probability ◽

10.1017/s0021900200019896 ◽

2001 ◽

Vol 38 (02) ◽

pp. 335-356 ◽

Cited By ~ 1

Author(s):

Mengxiang Tang ◽

Michael S. Waterman

Keyword(s):

Restriction Fragment ◽

Statistical Significance ◽

Poisson Approximation ◽

New Technique ◽

Restriction Maps ◽

Open Problems ◽

Dna Restriction ◽

A New Technique ◽

Stein Method ◽

Random Restriction

Optical mapping is a new technique to generate restriction maps of DNA easily and quickly. DNA restriction maps can be aligned by comparing corresponding restriction fragment lengths. To relate, organize, and analyse these maps it is necessary to rapidly compare maps. The issue of the statistical significance of approximately matching maps then becomes central, as in BLAST with sequence scoring. In this paper, we study the approximation to the distribution of counts of matched regions of specified length when comparing two DNA restriction maps. Distributional results are given to enable us to compute p-values and hence to determine whether or not the two restriction maps are related. The key tool used is the Chen-Stein method of Poisson approximation. Certain open problems are described.

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