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

This article presents a limit theorem for the gaps $\widehat{G}{i:n}:= X{n-i+1:n} - X_{n-i:n}$betweenorderstatistics$X_{1:n} \le \cdots \le X_{n:n}$ofasampleofsize$n$ 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)$hasanon-latticedistributionwithfinitemean$\mu_{\mbox{log}}$\mathrm{.}As$n\to \infty$thefinitedimensionaldistributionsofthegaps$\widehat{G}{i:n}$convergetothoseoflimitinggaps$G_i$ which are the numbers of points in a stationary renewal process with i.i.d. spacings $- \log(1 - H_j)$betweentimes$T{i-1}$and$T_i$ of births in a Yule process, that is $T_i:={\sum }_{k=1}^i{\epsilon }_k\mathrm{/}k$ for a sequence of i.i.d. exponential variables ${\epsilon }_k$ 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.