THE BERNOULLI SIEVE REVISITED

• 成果类型: Article
• 署名作者: Gnedin, Alexander V.; Iksanov, Alexander M.; Negadajlov, Pavlo; Roesler, Uwe
• 署名单位: Utrecht University; University of Kiel
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/08-AAP592
• 发表日期: 2009
• 页码: 1634-1655
• 关键词: regenerative compositions; asymptotic laws; LIMIT-THEOREMS; random-walks

摘要

We consider an occupancy scheme in which balls are identified with n points sampled from the standard exponential distribution, while the role of boxes is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behavior of five quantities: the index K-n* of the last occupied box, the number K-n of occupied boxes, the number K-n,K-0 of empty boxes whose index is at most K-n*, the index W-n of the first empty box and the number of balls Z(n) in the last occupied box. It is shown that the limiting distribution of properly scaled and centered K-n*, coincides with that of the number of renewals not exceeding log n. A similar result is shown for K-n and W-n under a side condition that prevents occurrence of very small boxes. The condition also ensures that K-n,K-0 converges in distribution. Limiting results for Z(n) are established under an assumption of regular variation.