ORDER-OF-MAGNITUDE BOUNDS FOR EXPECTATIONS INVOLVING QUADRATIC-FORMS

• 成果类型: Article
• 署名作者: DELAPENA, VH; KLASS, MJ
• 署名单位: Columbia University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176988740
• 发表日期: 1994
• 页码: 1044-1077
• 关键词: independent random-variables; decoupling inequalities; sums; SEQUENCES

摘要

Let X1, X2, ..., X(n) be independent mean-zero random variables and let a(ij), 1 less-than-or-equal-to i,j less-than-or-equal-or n, be an array of constants with a(ii) = 0. We present a method of obtaining the order of magnitude of EPHI(SIGMA1 less-than-or-equal-to i,j less-than-or-equal-to n a(ij)X(i)X(j)) for any such {X(i)} and {a(ij)} and any nonnegative symmetric (convex) function PHI with PHI(0) = 0 such that, for some integer k greater-than-or-equal-to 0, PHI(x2 - k) is convex and simultaneously PHI(x2-k-1) is concave on [0, infinity). The approximation is based on decoupling inequalities valid for all such mean-zero {X(i)} and reals {a(ij)} and a certain further 'independentization' procedure.