On the number of empty boxes in the Bernoulli sieve II

2011/10/17 by Alexander Iksanov, Iksanov, Alexander
Computer Science · Economics, Econometrics and Finance · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1110.3713

openalex publication_date 2011/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Bernoulli sieve is the infinite "balls-in-boxes" occupancy scheme with random frequencies P_k=W₁... W_k-1(1-W_k), where (W_k)_k∈\mn are independent copies of a random variable W taking values in (0,1). Assuming that the number of balls equals n, let L_n denote the number of empty boxes within the occupancy range. In the paper we investigate convergence in distribution of L_n in the two cases which remained open after the previous studies. In particular, provided that \me |log W|=\me |log (1-W)|=∞ and that the law of W assigns comparable masses to the neighborhoods of 0 and 1, it is shown that L_n weakly converges to a geometric law. This result is derived as a corollary to a more general assertion concerning the number of zero decrements of nonincreasing Markov chains. In the case that \me |log W|

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