Poisson Approximations for Sum of Bernoulli Random Variables and its Application to Ewens Sampling Formula

Abstract

The Ewens sampling formula is well-known as a distribution of a random partition of the set of integers {1, 2,. .. , n}. We give the condition that the number Kn of distinct components of the formula converges to the shifted Poisson distribution. Based on this convergence, we give the new approximations to the distribution of Kn, which are different from the approximations by Arratia et al. (2000, 2003). The formers are better than the latters. This is shown by comparing the bounds for the total variation distances between the distributions of the approximations and the distribution of Kn. Several examples are given to illustrate the results.

Recurrent mutation in the ancestry of a rare variant

“…For very large counts of the variant, the distribution of is well approximated by a Poisson with mean equal to the expected number of mutations per site on the gene genealogy, . This shifted-Poisson result is known already for the constant-size case ( Arratia et al 2000 ; Yamato 2017 ). In “A remark on the total number of mutations for large ” in the Appendix we argue that it should hold more generally.…”

“…The limiting large- distribution is Poisson, but shifted because there must be at least one mutation to produce copies, so it is that is Poisson. See Proposition 3.1 of Yamato (2017) . By the following heuristic argument, we suggest that this result holds more broadly, in particular for growing populations or ones in which decreases at least as fast with i as in the constant-size model, whatever the reason.…”

“…The limiting large- distribution is Poisson, but shifted because there must be at least one mutation to produce copies, so it is that is Poisson. See Proposition 3.1 of Yamato (2017) .…”

Shifted Binomial Approximation for the Ewens Sampling Formula (Ii)

“…The Poisson approximation to the distribution of the number K n of distinct components is studied by Arratia et al (2000) in detail with respect to the logarithmic combinatorial structure including the Ewens sampling formula. Differently from Arratia et al (2000), Yamato (2017b) approaches to the problem of Poisson approximation to L(K n ) by using the sum of independent Bernoulli random variables.…”

Shifted Binomial Approximations for Ewens Sampling Formula

“…The Poisson approximation to the distribution of the number K n of component are studied by Arratia et al (2000) in detail with respect to the logarithmic combinatorial structure including Ewens sampling formula. Differently from Arratia et al (2000), Yamato (2017) approaches to the problem of Poisson approximation to L(K n ) by using the sum of independent Bernoulli random variables.…”