On the number of empty boxes in the Bernoulli sieve II

Abstract

The Bernoulli sieve is the infinite "balls-in-boxes" occupancy scheme with random frequencies $P_k=W_1... W_{k-1}(1-W_k)$, where $(W_k)_{k\in\mn}$ 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 $\me |\log W|=\me |\log (1-W)|=\infty$ 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 $\me |\log W|<\infty$ and $\me |\log (1-W)|=\infty$ we derive several further possible modes of convergence in distribution of $L_n$. It turns out that the class of possible limiting laws for, properly normalized and centered, $L_n$ 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 in Mikosch and Resnick (2006).