An empirical process interpretation of a model of species survival

Abstract

We study a model of species survival recently proposed by Michael and Volkov. We interpret it as a variant of empirical processes, in which the sample size is random and when decreasing, samples of smallest numerical values are removed. Micheal and Volkov proved that the empirical distributions converge to the sample distribution conditioned not to be below a certain threshold. We prove a functional central limit theorem for the fluctuations. There exists a threshold above which the limit process is

A stochastic model for the evolution of species with random fitness

“…sequence with arbitrary distribution. Whence our results apply to the models in [2,6,9] (see Section 2.1). Besides, in the older papers the fitness is assigned uniformly while we use a general distribution µ.…”

“…Comparison with previous works. Our process extends those appeared in [2,6,9]. Aside from our general choice for the fitness law, the birth-and-death mechanism that we study is more general than those adopted in these papers.…”

“…The following theorem describes the long-term behaviour of a fixed fitness. Note that all f > f c belong to case (1), while all f < f c belong to (2). If f = f c , then case (2) applies if and only if (…”

“…The model has been generalized in [9] and [2]: there is still a toss of a coin to decide for creation or deletion, but instead of adding/removing one species at a time, increments are arbitrary random variables. Even with these assumptions, the same cut-off phenomenon of [6] appears.…”

“…In the original GMS, the lengths of subsequent births and deaths are geometrically distributed random variables (with parameters which sum up to 1) and in [2,9] they are geometrical convolutions of certain laws (where the parameters of these geometrically distributed number of convolutions, again, sum up to 1). In our model we group all subsequent creations and deletions: the length of subsequent creations {X n } n∈N and the length of subsequent annihilations {Y n } n∈N are such that {(X n , Y n )} n∈N is an i.i.d.…”