On the lower tail probabilities of some random series

• 成果类型: Article
• 署名作者: Lifshits, MA
• 署名单位: Universites de Strasbourg Etablissements Associes; Universite de Strasbourg
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 424-442
• 关键词: gaussian seminorms

摘要

The behavior of tail probabilities P{S less than or equal to r}, r --> 0 is investigated, where S is a series S = Sigma lambda(j)Z(j) generated by some sequence of positive numbers {lambda(j)} and by a sequence {Z(j)} of independent copies of a positive random variable Z. We present the exact asymptotic expression for P{S less than or equal to r} by means of Laplace transform Lambda(gamma) = E exp{-gamma S} under weak assumptions on the behavior of the tail probabilities of Z in the vicinity of zero. The bounds of accuracy are also given, and under weak supplementary smoothness conditions the asymptotic properties of the density of S are investigated.