On the number of empty boxes in the Bernoulli sieve II

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:122

Article

Iksanov, Alexander

关键词: Bernoulli sieve;    Continuous mapping theorem;    Convergence in distribution;    Depoissonization;    Infinite occupancy scheme;    Renewal shot-noise process;   

DOI  :  10.1016/j.spa.2012.04.010

来源: Elsevier

【 摘 要 】

The Bernoulli sieve is the infinite balls-in-boxes occupancy scheme with random frequencies P-k = W-1 center dot center dot center dot Wk-1(1 - W-k), where (W-k)(k is an element of N) are independent copies of a random variable W taking values in (0, I). Assuming that the number of balls equals n, let L-n denote the number of empty boxes within the occupancy range. In this paper, we investigate convergence in distribution of L-n in the two cases which remained open after the previous studies. In particular, provided that E vertical bar log W vertical bar = E vertical bar log(1 - W)vertical bar = infinity and that the law of W assigns comparable masses to the neighborhoods of 0 and I, 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 E vertical bar log W vertical bar < infinity and E vertical bar log(1 - W)vertical bar = infinity, we derive several further possible modes of convergence in distribution of L-n. It turns out that, the class of possible limiting laws for L-n, properly normalized and centered, includes normal laws and spectrally negative stable laws with finite mean. While investigating the second problem, we develop some general results concerning the weak convergence of renewal shot-noise processes. This allows us to answer a question asked by Mikosch and Resnick (2006) [18]. (C) 2012 Elsevier By. All rights reserved.

【 授权许可】

Free