Improved compound Poisson approximation for the number of occurrences of any rare word family in a stationary markov chain

Abstract

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.

Natural and artificial RNAs occupy the same restricted region of sequence space

“…General error-bounds for this approximation are known only for motifs consisting of a single module as long as no version of the motif is a proper sub-word of another version of it (Roquain and Schbath 2007); in particular, bounds for the error are unknown for motifs with two or more modules with unbounded gaps. However, due to Markov's inequality (Durrett 2004), for an arbitrary motif m we have P[W $ 1] # E(W), where E(W) denotes the expected value of W. But, due to the stationarity of our probabilistic model and the linearity of the expectation operator, E(W) = Np, and hence P[W $ 1] # Np.…”

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).…”

Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences

“…See also Lladser et al [16]. Approximations using the normal distribution and the Poisson distribution are in Schbath [31], Nicodème et al [21], Reinert et al [28] and Roquain and Schbath [30]. Large deviations asymptotics are in Régnier and Szpankowski [27], Régnier [25] and Régnier and Denise [26].…”

“…The leading term of the asymptotics of g a (n, it) when t is close to the origin is due to Mann [17] (pp. [24][25][26][27][28][29][30][31][32][33], and is given in Proposition 4.2 below.…”