2016/01/17 by Gerold Alsmeyer, Alsmeyer, Gerold, Alexander Iksanov +3
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60C05 #msc:60F17 #primary 60F17 #secondary 60C05
paper · pdf · doi:10.48550/arxiv.1601.04274
22 pages
arxiv created 2016/01/17 · arxiv updated 2016/01/19
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 values 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→∞.