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

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).