Occupancy Schemes Associated to Yule Processes

TL;DR

This paper studies an occupancy problem linked to Yule processes, analyzing the asymptotic distribution of empty bins and the impact of a random environment, revealing the role of rare events in the model's behavior.

Contribution

It introduces a novel analysis of occupancy schemes associated with Yule processes, including convergence results and the influence of a complex random environment.

Findings

Convergence of empty bin distributions to mixed Poisson processes.

Impact of the random environment components on the model's behavior.

Identification of rare events affecting empty bin counts.

Abstract

An occupancy problem with an infinite number of bins and a random probability vector for the locations of the balls is considered. The respective sizes of bins are related to the split times of a Yule process. The asymptotic behavior of the landscape of first empty bins, i.e., the set of corresponding indices represented by point processes, is analyzed and convergences in distribution to mixed Poisson processes are established. Additionally, the influence of the random environment, the random probability vector, is analyzed. It is represented by two main components: an i.i.d. sequence and a fixed random variable. Each of these components has a specific impact on the qualitative behavior of the stochastic model. It is shown in particular that for some values of the parameters, some rare events, which are identified, play an important role on average values of the number of empty bins in some regions.