An Empirical Process Interpretation of a Model of Species Survival

TL;DR

This paper analyzes a species survival model through the lens of empirical process theory, proving a functional central limit theorem and characterizing the limit process's Gaussian properties at different thresholds.

Contribution

It provides a novel empirical process interpretation of the species survival model and establishes a functional central limit theorem for its fluctuations.

Findings

Empirical distributions converge to a conditioned sample distribution.

A functional central limit theorem is proved for the model.

The limit process is Gaussian above a threshold and half-Gaussian at the threshold.

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 Gaussian with variance bounded below by a positive constant, while at the threshold it is half-Gaussian.