Gaps and interleaving of point processes in sampling from a residual allocation model. (English) Zbl 1428.62190

Summary: 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 A. Gnedin et al. for the Bernoulli sieve [in: Fifth colloquium on mathematics and computer science. Lectures from the colloquium. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). 235–242 (2008; Zbl 1357.60096)].


60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
60K05 Renewal theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)


Zbl 1357.60096
Full Text: DOI arXiv Euclid


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