On extremal distributions and sharp Lp-bounds for sums of multilinear forms

• 成果类型: Article
• 署名作者: De La Peña, VH; Ibragimov, R; Sharakhmetov, S
• 署名单位: Columbia University; Yale University; Tashkent State University of Economics
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 630-675
• 关键词: decoupling inequalities; LIMIT-THEOREMS; khintchine; moments; DOMINATION; constants

摘要

In this paper we present a study of the problem of approximating the expectations of functions of statistics in independent and dependent random variables in terms of the expectations of functions of the component random variables. We present results providing sharp analogues of the Burkholder-Rosenthal inequalities and related estimates for the expectations of functions of sums of dependent nonnegative r.v.'s and conditionally symmetric martingale differences with bounded conditional moments as well as for sums of multilinear forms. Among others, we obtain the following sharp inequalities: E(Sigma(k=1)(n) X-k)(t) less than or equal to 2 max(Sigma(k=1)(n) EXkt, (Sigma(k=1)(n) a(k))(t)) for all nonnegative r.v.'s X-1,...,X-n with E(X-k \ X-1,...,Xk-1) less than or equal to a(k), EXk1 < infinity, k = 1,...,n, 1 < t < 2; E(Sigma(k=1)(n)X(k))(t) less than or equal to Etheta(t) (1) max(Sigma(k=1)(n) b(k), (Sigma(k=1)(n) a(k)(s))(t/s)) for all nonnegative r.v.'s X-1,...,X-n with E(X-k(s) \ X-1,...,Xk-1) less than or equal to a(k)(s), E(X-k(t) \ X-1,...Xk-1) less than or equal to b(k). k=1,...,n, 1 < t < 2, 0 < s less than or equal to t-1 or tless than or equal to2, 0 < s less than or equal to 1, where theta(1) is a Poisson random variable with parameter 1. As applications, new decoupling inequalities for sums of multilinear forms are presented and sharp Khintchine-Marcinkiewicz-Zygmund inequalities for generalized moving averages are obtained. The results can also be used in the study of a wide class of nonlinear statistics connected to problems of long-range dependence and in an econometric setup, in particular, in stabilization policy problems and in the study of properties of moving average and autocorrelation processes. The results are based on the iteration of a series of key lemmas that capture the essential extremal properties of the moments of the statistics involved.