Bounds on the tail probability of 𝑈-statistics and quadratic forms

作者：Víctor Peña, Stephen Montgomery-Smith · 发表于：Bulletin of the American Mathematical Society · 年份：1994 · DOI：10.1090/s0273-0979-1994-00522-1 · 被引用次数：30 · 研究领域：Advanced Banach Space Theory、Functional Equations Stability Results、Mathematical and Theoretical Analysis

It is very common for expressions of the form: / , fiv-kWf,, ■■■ , X¡k)to appear in probability theory.Here {X¡} is a sequence of independent random variables taking values in a measurable space (S,<9*), and {f¡¡.-ik} is a sequence of measurable functions from Sk into a Banach Space (B, || • ||).Special cases of this type of random variable appear, for example, in statistics in the form of U-statistics and quadratic forms.Throughout we will refer to them as generalized U-statistics.There is great interest in decoupling such quantities, that is, in replacing the above quantity by the expressionwhere {X\l)}, {x\2)},... , {X\k)} are k independent copies of {X¡}.Decoupling inequalities allows one to compare expressions of the first kind with expressions of the second kind.Such results permit the almost-direct transfer of results for sums of independent random variables to the case of generalized U-statistics.The reason for this is that, conditionally on {xf^}, ..., {XJ^}, the second sum above is a sum of independent random variables.It is important to remark that such results have led to the development of several optimal results in the functional theory of U-statistics (cf.[1] and [7]) and various other areas, including the study of the invertibility of large matrices (cf.[2]), stochastic integration (cf.[10]), and the study of integral operators on Lebesgue-Bochner spaces (cf. a result of T. R. McConnell and D. Burkholder found in [3]).Aside from those directly cited in this pa...