A Multivariate Generalization of Hoeffding’s Inequality
نویسنده
چکیده
In the study of U -statistics we need a multivariate version of this result. The goal of this paper is to present such an inequality. To formulate it first we have to introduce some notations. Let us fix a positive integer k and some real numbers a(j1, . . . , jk) for all sets of arguments {j1, . . . , jk} such that 1 ≤ jl ≤ n, 1 ≤ l ≤ k, and jl 6= jl′ if l 6= l, in such a way that the numbers a(j1, . . . , jk) are symmetric functions of their arguments, i.e. a(j1, . . . , jk) = a(jπ(1), . . . , jπ(k)) for all permutations π ∈ Πk of the set {1, . . . , k}. Let us define with the help of the above real numbers and a sequence of independent random variables ε1, . . . , εn, P (εj = 1) = P (εj = −1) = 12 , 1 ≤ j ≤ n, the random variable Z = ∑
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