Poisson approximations for runs and patterns of rare events

Abstract

Consider a sequence of Bernoulli trials with success probability p, and let Nn,k denote the number of success runs of length among the first n trials. The Stein–Chen method is employed to obtain a total variation upper bound for the rate of convergence of Nn,k to a Poisson random variable under the standard condition npk→λ. This bound is of the same order, O(p), as the best known for the case k = 1, i.e. for the classical binomial-Poisson approximation. Analogous results are obtained for occurrences of word patterns, where, depending on the nature of the word, the corresponding rate is at most O(pk–m) for some m = 0, 2, ···, k – 1. The technique is adapted for use with two-state Markov chains. Applications to reliability systems and tests for randomness are discussed.

Compound Poisson approximation of word counts in DNA sequences

“…variables X i has been widely considered. The convergence of N(W) for rare words to a Poisson variable is then proved either with generating functions or by using the Chen-Stein method Papastavridis (1988a, 1988b), Godbole (1991), Hirano and Aki (1993), Godbole and Scha ner (1993), Fu (1993)). When the sequence (X i ) i=1 n is a rst order Markov c hain on f0 1g and W is a run of ones, some of these authors show the convergence of N(W) t o a P oisson or compound Poisson variable when h n !…”

Significance Score of Motifs in Biological Sequences

“…We can basically classify these approximations in three categories: 1) Gaussian approximations (Cowan, 1991;Kleffe & Borodovski, 1997;Nuel, 2010;Pevzner et al, 1989;Prum et al, 1995); 2) Poisson approximations Erhardsson (2000); Geske et al (1995); Godbole (1991); Reinert & Schbath (1999); Roquain & Schbath (2007); 3) large deviations approximations Denise et al (2001);Nuel (2004). In this chapter we deliberately left aside the Poisson-based approximations and considered only two of these approximations: the (Near-) Gaussian approximations with NG h (n), and the large deviations based approximations with CB(n) and BR(n).…”

Approximating the distributions of runs and patterns

“…The primary tool used to obtain μ n and the bound ε n is the Stein-Chen method (Chen 1975), and this method has been refined by various authors Arratia et al (1990), Barbour and Eagleson (1983), Barbour and Eagleson (1984), Barbour and Eagleson (1987), Barbour and Hall (1984), Godbole (1990a), Godbole (1990b), Godbole (1991), Godbole and Schaffner (1993), and Holst et al (1988). This method has also been extended to compound Poisson approximations for the distributions of runs and patterns and Barbour and Chryssaphinou (2001) provides an excellent theoretical review of these approximations.…”