Strong law of large numbers for multilinear forms

• 成果类型: Article
• 署名作者: Gadidov, A
• 署名单位: Gannon University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1022855655
• 发表日期: 1998
• 页码: 902-923
• 关键词: random-variables; inequalities; sums

摘要

Let m greater than or equal to 2 be a nonnegative integer and let {X-(l), X-i((l))}(i) (is an element of) (N), l = 1,..., m, be m independent sequences of independent and identically distributed symmetric random variables. Define S-n = Sigma(1 less than or equal to 1, ... i) (m less than or equal to n) X-i1((1)) ... X-im((m)), and let {gamma(n)}(n is an element of N) be a nondecreasing sequence of positive numbers, tending to infinity and satisfying some regularity conditions. For m = 2 necessary and sufficient conditions are obtained for the strong law of large numbers gamma(n-1)S(n) --> 0 a.s. to hold, and for m > 2 the strong law of large numbers is obtained under a condition on the growth of the truncated variance of the X-(l).