Further Applications of Decoupling

Abstract

In this chapter we present several additional applications of decoupling. The first section gives several results for randomly stopped U-statistics, including an extension of Wald’s equations. This result contains as a special case Wald’s second equation for sums and is the basis for extensions of sequential methods to U-statistics. We also develop several bounds (some of which are shown to be sharp) on the Lp-norm of randomly stopped U-statistics and apply these results to prove convergence of moments in Anscombe’s theorem for U-statistics. The key tools for the second part of Section 8.1 consist of bounds on the moments of U-statistics including an extension of Levy’s inequality for the La-norm of sums of independent mean zero random variables to the case of U-statistics with degenerate kernels (already encountered, albeit implicitly, in Chapter 4). This extension of Levy’s inequality shows that, in certain aspects, U--statistics behave more like sums of independent random variables than like martingales (in other aspects like, e.g., regarding the law of large numbers, converse part--see Section 4.1—, U-statistics do not behave at all as sums of independent variables).