Functional limit theorems for the number of occupied boxes in the Bernoulli sieve

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:127

Article

Alsmeyer, Gerold¹  Iksanov, Alexander²  Marynych, Alexander^1,2 

[1] Univ Munster, Dept Math & Comp Sci, Inst Stat Math, Orleans Ring 10, D-48149 Munster, Germany

[2] Taras Shevchenko Natl Univ Kyiv, Fac Cybernet, UA-01601 Kiev, Ukraine

关键词: Bernoulli sieve;    Infinite urn model;    Perturbed random walk;    Renewal theory;   

DOI  :  10.1016/j.spa.2016.07.007

来源: Elsevier

【 摘 要 】

The Bernoulli sieve is the infinite Karlin balls-in-boxes scheme with random probabilities of stick breaking type. Assuming that the number of placed balls equals n, we prove several functional limit theorems (FLTs) in the Skorohod space D [0, 1] endowed with the J(1)- or M-1-topology for the number K-n* (t) of boxes containing at most [n(t)] balls, t is an element of [0, 1], and the random distribution function K-n*(t)/K-n*(1), as n -> infinity. The limit processes for K-n*(t) are of the form (X(1) - X((1 - t)-))t is an element of [0,1], where X is either a Brownian motion, a spectrally negative stable Levy process, or an inverse stable subordinator. The small value probabilities for the stick-breaking factor determine which of the alternatives occurs. If the logarithm of this factor is integrable, the limit process for K-n*(t)/K-n*(1) is a Levy bridge. Our approach relies upon two novel ingredients and particularly enables us to dispense with a Poissonization-de-Poissonization step which has been an essential component in all the previous studies of K-n*(1). First, for any Karlin occupancy scheme with deterministic probabilities (Pk)k >= 1, we obtain an approximation, uniformly in t is an element of [0, 1], of the number of boxes with at most [n(t)] balls by a counting function defined in terms of (Pk)k >= 1. Second, we prove several FLTs for the number of visits to the interval [0, nt] by a perturbed random walk, as n -> infinity. If the stick-breaking factor has a beta distribution with parameters theta > 0 and 1, the process K-n*(t))t is an element of [0,1] has the same distribution as a similar process defined by the number of cycles of length at most [nt] in a theta-biased random permutation a.k.a. a Ewens permutation with parameter theta. As a consequence, our FLT with Brownian limit forms a generalization of a FLT obtained earlier in the context of Ewens permutations by DeLaurentis and Pittel (1985), Hansen (1990), Donnelly et al. (1991), and Arratia and Tavard (1992). (C) 2016 Elsevier B.V. All rights reserved.

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