Bounded size bias coupling: a Gamma function bound, and universal Dickman-function behavior

• 成果类型: Article
• 署名作者: Arratia, Richard; Baxendale, Peter
• 署名单位: University of Southern California
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0572-x
• 发表日期: 2015
• 页码: 411-429
• 关键词: steins method; PROBABILITY-MEASURES

摘要

Under the assumption that the distribution of a nonnegative random variable admits a bounded coupling with its size biased version, we prove simple and strong concentration bounds. In particular the upper tail probability is shown to decay at least as fast as the reciprocal of a Gamma function, guaranteeing a moment generating function that converges everywhere. The class of infinitely divisible distributions with finite mean, whose L,vy measure is supported on an interval contained in for some , forms a special case in which this upper bound is logarithmically sharp. In particular the asymptotic estimate for the Dickman function, that for large , is shown to be universal for this class. A special case of our bounds arises when is a sum of independent random variables, each admitting a 1-bounded size bias coupling. In this case, our bounds are comparable to Chernoff-Hoeffding bounds; however, ours are broader in scope, sharper for the upper tail, and equal for the lower tail. We discuss bounded and monotone couplings, give a sandwich principle, and show how this gives an easy conceptual proof that any finite positive mean sum of independent Bernoulli random variables admits a 1-bounded coupling with the same conditioned to be nonzero.