A normal comparison inequality and its applications

• 成果类型: Article
• 署名作者: Li, WBV; Shao, QM
• 署名单位: University of Delaware; University of Oregon
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400100176
• 发表日期: 2002
• 页码: 494-508
• 关键词: gaussian-processes; iterated logarithm; brownian-motion; exit times; cones; LAW

摘要

Let xi = (xi(i), 1 less than or equal to i less than or equal to n) and eta = (eta(t), 1 less than or equal to i less than or equal to n) be standard normal random variables with covariance matrices R-1 = (r(j,i)(1)) and R-0 = (r(j,i)(0)), resplectively, Slepian's lemma says that if r(j,i)(1) greater than or equal to r(j,i)(0) for 1 less than or equal to i, j less than or equal to n, the lower bound P (xi(t) < u for 1 <1 < n)/P(eta(i) less than or equal to u for 1 less than or equal to i less than or equal to n) is at least 1. In this paper an upper bound is given. The usefulness of the upper bound is justified with three concrete applications: (i) the new law of the iterated logarithm of Erdos and Revesz, (ii) the probability that a random polynomial does not have a real zero and (iii) the rando pursuit problem for fractional Brownian particles. In particular, a conjecture of Kesten(1992) on the random pursuit problem for Brownian particles is confirmed, which leads to estimates of principal eigenvalues.