A symmetrization-desymmetrization procedure for uniformly good approximation of expectations involving arbitrary sums of 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; Lund University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1019160512
• 发表日期: 2000
• 页码: 1884-1907
• 关键词: random bilinear-forms; magnitude bounds; delta(2)-functions; ORDER

摘要

Let Phi be a symmetric function, nondecreasing on [0, infinity) and satisfying a Delta (2) growth condition, (X-1, Y-1), (X-2, Y-2),...,(X-n, Y-n) be independent random vectors such that (for each 1 less than or equal to i less than or equal to n) either Y-i = X-i or Y-i is independent of all the other variates, and the marginal distributions of {X-i} and {Y-j} are otherwise arbitrary. Let {f(ij)(x, y)}(1 less than or equal toi, j less than or equal ton) be any array of real valued measurable functions. We present a method of obtaining the order of magnitude of E Phi(Sigma (1 less than or equal toi, j less than or equal ton) f(ij)(X-i,Y-j)). The proof employs a double symmetrization, introducing independent copies {(X) over tildei, (Y) over tilde (j)} of {X-i, Y-j}, and moving from summands of the form f(ij)(X-i. Y-j) to what we call f(ij)((s))(X-i, Y-j, (X) over tilde (i), (Y) over tilde (j)). Substitution of fixed constants (x) over tilde (i) and (y) over tilde (j) far (X) over tilde (i) and (Y) over tilde (j) results in f(ij)((s))(X-i, Y-j, (x) over tilde (i), (y) over tilde (j)), which equals f(ij)(X-i, Y-j) adjusted by a sum of quantities of first order separately in X-i and Y-j. Introducing further explicit first-order adjustments, call them g(1ij)(X-i, (x) over tilde, (y) over tilde) and g(2ij)(Y-j, (x) over tilde, (y) over tilde), it is proved that E Phi(Sigma (1 less than or equal toi, j less than or equal ton)(X-i, Y-j, (x) over tilde (i), (y) over tilde (j)) - g(1ij)(X-i, (x) over tilde, (y) over tilde) - g(2ij)(Y-j, (x) over tilde, (y) over tilde))) less than or equal to (alpha) E Phi(root Sigma (1 less than or equal toi, j less than or equal ton) (f(ij)((s))(X-i, Y-j, (x) over tilde (i), (y) over tilde (j)))(2)) approximate to (alpha) Phi (f((s)), X, Y, (x) over tilde, (y) over tilde) where the latter is an explicitly computable quantity. For any (x) over tilde (0) and (y) over tilde (0) which come within a factor of two of minimizing Phi (f((s)), X, Y, (x) over tilde, (y) over tilde) it is shown that E Phi(Sigma (1 less than or equal toi,j less than or equal ton) f(ij)(X-i, Y-j)) approximate to (alpha) max {Phi (f((s)), X, Y, (x) over tilde (0), (y) over tilde (0)), E Phi(Sigma (1 less than or equal toi, j less than or equal ton) (f(ij)(X-i, (y) over tilde (0)(j)) + f(ij)((x) over tilde (0)(j), Y-j) - f(ij)((x) over tilde (0)(i), (y) over tilde (0)(j)) + g(1ij)(X-i, (x) over tilde (0)(i), (y) over tilde (0)(j)) + g(2ij)(Y-j, (x) over tilde (0)(i), (y) over tilde (0)(j))))}. which is computable (approximable) in terms of the underlying random variables. These results extend to the expectation of Phi of a sum of functions of k-components.