Order of magnitude bounds for expectations of Δ2-functions of nonnegative random bilinear forms and generalized U-statistics

• 成果类型: Article
• 署名作者: Klass, MJ; Nowicki, K
• 署名单位: University of California System; University of California Berkeley; University of California System; University of California Berkeley; Lund University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 1471-1501
• 关键词: random-variables; decoupling inequalities; quadratic-forms; sums

摘要

Let X-1, Y-1, Y-2,..., X-n, Y-n be independent nonnegative rv's and let {b(ij)}(1 less than or equal to i, j less than or equal to n) be an array of nonnegative constants. We present a method of obtaining the order of magnitude of E Phi (Sigma(1 less than or equal to i, j less than or equal to n) b(ij) X-i Y-j), for any such {X-i}, {Y-i} and {b(ij)} and any nondecreasing function Phi on [0, infinity) with Phi(0) = 0 and satisfying a Delta(2) growth condition Furthermore, this technique is extended to provide the order of magnitude of E Phi (Sigma(1 less than or equal to i, j less than or equal to n) b(ij) X-i Y-j), where {f(ij)(x, y)}(1 less than or equal to i, j less than or equal to n) is any array of nonnegative functions. Far arbitrary functions {g(ij)(x,y)}(1 less than or equal to i not equal j less than or equal to n), the aforementioned approximation enables us to identify the order of magnitude of E Phi (/Sigma (1 less than or equal to i not equal j less than or equal to n) g(ij)(X-i, X-j)/) whenever decoupling results and Khintchine-type inequalities apply, such as Phi is convex, L(g(ij)(X-i, X-j)) = L(g(ji)(X-j, X-i)) and Eg(ij)(X-i, x) = 0 for all x in the range of X-j.