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

Abstract

This article presents a limit theorem for the gaps ${\hat{G}}_{i:n}:=X_{n-i+1:n}-X_{n-i:n}$ 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 $(P_1,P_2,\dots )$ 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 $μ_{\mbox{log}}$. As $n\to \infty$ the finite dimensional distributions of the gaps ${\hat{G}}_{i:n}$ 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 $T_i:={\sum }_{k=1}^i{\epsilon }_k/k$ for a sequence of i.i.d. exponential variables ${\epsilon }_k$ with mean 1. A consequence is that the mean of ${\hat{G}}_{i:n}$ converges to the mean of $G_i$, which is $1/(i μ_{\mbox{log}} )$. This limit theorem simplifies and extends a result of Gnedin, Iksanov and Roesler for the Bernoulli sieve.

26 pages