On the number of empty boxes in the Bernoulli sieve. II. (Q432511)

The Bernoulli sieve considered in this paper is an infinite urn model with random frequencies \(P_1,P_2,\ldots\) in which (abstract) balls are allocated over an infinite sequence of (abstract) urns \(1,2,\ldots\) independently conditionally given \((P_k)_{k\geq1}\) with probability \(P_j\) of hitting urn \(j\), and where \(P_k=W_1W_2\ldots W_{k-1}(1-W_k)\) for independent copies \(W_1,W_2,\ldots\) of a random variable \(W\) taking values in \((0,1)\). For \(n\) balls, let \(K_n\) denote the number of occupied urns and \(M_n\) the largest index of the occupied urns. Then \(L_n=M_n-K_n\) is the number of empty urns within the occupancy range. If \(\text{E}|\log W|=\text{E}|\log(1-W)|=\infty\) and the distribution of \(W\) assigns comparable masses to the neighborhoods of \(0\) and \(1\), it is shown that \(L_n\) converges in distribution to a geometric law. If \(\text{E}|\log W|<\infty\) and \(\text{E}|\log(1-W)|=\infty\), several further modes of convergence in distribution of \(L_n\) are derived. These results complement previous studies of the asymptotic distribution of \(L_n\).NEWLINENEWLINEFor part I, see [Stochastics, accepted (\url{doi:10.1080/17442508.2012.688974})].