Exact inequalities for sums of asymmetric random variables, with applications

• 成果类型: Article
• 署名作者: Pinelis, Iosif
• 署名单位: Michigan Technological University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0055-4
• 发表日期: 2007
• 页码: 605-635
• 关键词: linear-combinations; distributions; bounds

摘要

Let BS1,..., BSn be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter p is an element of (0,1). Let m(*)(p) := (1 + p + 2p(2))/(2 root p - p(2) + 4p(2)) if 0 < p <= 1/2 and m(*)(p) := 1 if 1/2 <= p < 1. Let m >= m(*)(p). Let f be such a function that f and f '' are nondecreasing and convex. Then it is proved that for all nonnegative numbers c(1),..., c(n) one has the inequality Ef(c(1)BS(1) + center dot center dot center dot + c(n)BS(n)) <= Ef(s((m))(BS1 + center dot center dot center dot + BSn)), where s(m) := (1/n Sigma(n)(i=1) c(i)(2m))(1/2m). The lower bound m*(p) on m is exact for each p is an element of (0,1). Moreover, Ef(c(1)BS(1) + center dot center dot center dot + c(n)BS(n)) is Schur-concave in (c(1)(2m),..., c(n)(2m)). A number of corollaries are obtained, including upper bounds on generalized moments and tail probabilities of (super) martingales with differences of bounded asymmetry, and also upper bounds on the maximal function of such (super) martingales. Applications to generalized self-normalized sums and t-statistics are given.