On the number of empty boxes in the Bernoulli sieve II

The Bernoulli sieve is the infinite “balls-in-boxes” occupancy scheme with random frequencies Pk=W1⋯Wk−1(1−Wk)Pk=W1⋯Wk−1(1−Wk), where (Wk)k∈N(Wk)k∈N are independent copies of a random variable WW taking values in (0,1)(0,1). Assuming that the number of balls equals nn, let LnLn denote the number of empty boxes within the occupancy range. In this paper, we investigate convergence in distribution of LnLn in the two cases which remained open after the previous studies. In particular, provided that E|logW|=E|log(1−W)|=∞E|logW|=E|log(1−W)|=∞ and that the law of WW assigns comparable masses to the neighborhoods of 0 and 1, it is shown that LnLn 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|logW|<∞E|logW|<∞ and E|log(1−W)|=∞E|log(1−W)|=∞, we derive several further possible modes of convergence in distribution of LnLn. It turns out that the class of possible limiting laws for LnLn, 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].