关于伯努利筛子中的空箱数
On the number of empty boxes in the Bernoulli sieve
摘要：
伯努利筛是具有随机频率$P_k=W_1...W_{k-1}(1-W_k)$的无限“盒中球”占用方案，其中$(W_k)_{k\in\mn}$是随机变量$W$的独立副本，其取值为$(0,1)$。假设球的数量等于$n$，令$L_n$表示占用范围内的空盒子的数量。本文证明了在正则变分假设下，$L_n$，适当正规化无中心，弱收敛到一个反稳定从属泛函。证明依赖于$(\log P_k)$是扰动随机游动的观察。特别地，对一般扰动随机游动得到了一些结果。另一个结果的文件指出，每当$L_n$弱收敛（没有归一化）的限制法律是混合泊松。
英文摘要：
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. The paper proves that, under a regular variation assumption, $L_n$, properly normalized without centering, weakly converges to a functional of an inverse stable subordinator. Proofs rely upon the observation that $(\log P_k)$ is a perturbed random walk. In particular, some results for general perturbed random walks are derived. The other result of the paper states that whenever $L_n$ weakly converges (without normalization) the limiting law is mixed Poisson.
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