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

Gerold Alsmeyer, Alexander Iksanov and Alexander Marynych

Stochastic Processes and their Applications, 2017, vol. 127, issue 3, 995-1017

Abstract: 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 J1- or M1-topology for the number Kn∗(t) of boxes containing at most [nt] balls, t∈[0,1], and the random distribution function Kn∗(t)/Kn∗(1), as n→∞. The limit processes for Kn∗(t) are of the form (X(1)−X((1−t)−))t∈[0,1], where X is either a Brownian motion, a spectrally negative stable Lévy 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 Kn∗(t)/Kn∗(1) is a Lévy 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 Kn∗(1). First, for any Karlin occupancy scheme with deterministic probabilities (pk)k≥1, we obtain an approximation, uniformly in t∈[0,1], of the number of boxes with at most [nt] 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→∞. If the stick-breaking factor has a beta distribution with parameters θ>0 and 1, the process (Kn∗(t))t∈[0,1] has the same distribution as a similar process defined by the number of cycles of length at most [nt] in a θ-biased random permutation a.k.a. a Ewens permutation with parameter θ. 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 Tavaré (1992).

Keywords: Bernoulli sieve; Infinite urn model; Perturbed random walk; Renewal theory (search for similar items in EconPapers)
Date: 2017
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S030441491630120X
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:127:y:2017:i:3:p:995-1017

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2016.07.007

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().