## 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)].

### MSC:

 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
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### References:

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