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

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 $\mu_{\log}$. 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 $1/(i\mu_{\log})$. This limit theorem simplifies and extends a result of Gnedin, Iksanov and Roesler for the Bernoulli sieve.