Gaps and interleaving of point processes in sampling from a residual allocation model

TL;DR

This paper establishes a limit theorem for the gaps between order statistics in samples from a residual allocation model, linking them to a stationary renewal process and extending previous results for the Bernoulli sieve.

Contribution

It introduces a new limit theorem describing the asymptotic behavior of gaps in residual allocation models, generalizing prior work on the Bernoulli sieve.

Findings

Gaps converge to those of a stationary renewal process.

Mean of gaps converges to 1/(i * μ_log).

Extends results of Gnedin, Iksanov, and Roesler.

Abstract

This article presents a limit theorem for the gaps $\widehat{G}_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$ between order statistics $X_{1:n} \le \cdots \le X_{n:n}$ of a sample of size $n$ from a random discrete distribution on the positive integers $(P_1, P_2, \ldots)$ governed by a residual allocation model (also called a Bernoulli sieve) $P_j:= H_j \prod_{i=1}^{j-1}(1-H_i)$ for a sequence of independent random hazard variables $H_i$ which are identically distributed according to some distribution of $H \in (0,1)$ such that $- \log(1 - H)$ has a non-lattice distribution with finite mean $\mu_{\mbox{log}}$. As $n\to \infty$ the finite dimensional distributions of the gaps $\widehat{G}_{i:n}$ converge to those of limiting gaps $G_i$ which are the numbers of points in a stationary renewal process with i.i.d. spacings $- \log(1 - H_j)$ between times $T_{i-1}$ and $T_i$ of births in a Yule process, that is $T_i := \sum_{k=1}^i \varepsilon_{k}/k$ for a sequence of i.i.d. exponential variables $\varepsilon_k$ with mean 1. A consequence is that the mean of $\widehat{G}_{i:n}$ converges to the mean of $G_i$, which is $1/(i \mu_{\mbox{log}} )$. This limit theorem simplifies and extends a result of Gnedin, Iksanov and Roesler for the Bernoulli sieve.