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

Abstract

This article presents a limit theorem for the gaps Gi:n := Xn−i+1:n − Xn−i:n between order statistics X1:n ≤ · · · ≤ Xn:n of a sample of size n from a random discrete distribution on the positive integers (P1, P2, . . .) governed by a residual allocation model (also called a Bernoulli sieve) Pj := Hj j−1 i=1 (1−Hi) for a sequence of independent random hazard variables Hi which are identically distributed according to some distribution of H ∈ (0, 1) such that − log(1 − H) has a non-lattice distribution with finite mean µ log . As n → ∞ the finite dimensional distributions of the gaps Gi:n converge to those of limiting gaps Gi which are the numbers of points in a stationary renewal process with i.i.d. spacings − log(1 − Hj) between times Ti−1 and Ti of births in a Yule process, that is Ti := i k=1 ε k /k for a sequence of i.i.d. exponential variables ε k with mean 1. A consequence is that the mean of Gi:n converges to the mean of Gi, which is 1/(iµ log ). This limit theorem simplifies and extends a result of Gnedin, Iksanov and Roesler for the Bernoulli sieve.

Regenerative random permutations of integers

“…These simulation results (7.1) are explained by the following lemma, which provides an alternative approach to the evaluation of u k derived from a RAM. This lemma is suggested by work of Gnedin and coauthors on the Bernoulli sieve [49,44,43], and following work on extremes and gaps in sampling from a RAM by Pitman and Yakubovich [86,82]. = W for some distribution of W on (0, 1).…”

“…A key ingredient in this evaluation is the fact that in the GEM(θ) model the random occupation times G j of Q are independent geometric variables, see [86,82]. For a more general RAM, the G j 's may not be independent, and they may not be exactly geometric, only conditionally so given G j ≥ 1.…”

Renewal sequences and record chains related to multiple zeta sums

“…The key to our analysis is the Markov chain ( Q k ) given by the following lemma from [51, Lemma 7.1]. This lemma is suggested by work of Gnedin and coauthors on the Bernoulli sieve [30,31], and subsequent work on extremes and gaps in sampling from a RAM by Pitman and Yakubovich [49,52].…”

Renewal theory for iterated perturbed random walks on a general branching process tree: intermediate generations

“…A survey of various results for PRWs T (with not necessarily positive ξ and η) and, in particular, counterparts of some renewal-theoretic results can be found in the book [14]. An incomplete list of more recent papers addressing various aspects of the PRWs includes [1], [9], [17], [22], [23], and [24]. We proceed by recalling the construction of a general branching process (a.k.a.…”