Moments of randomly stopped \(U\)-statistics
Let \(U_n\) be a \(U\)-statistics of the general form \(U_n=\sum_{1\leq i_1<\dots< i_k\leq n} f(X_{i_1},\dots,X_{i_k})\), where \(\{X_n\}\) is a sequence of i.i.d.\ random variables. Let \(T\) be a stopping time with respect to the filtration \(\{F_n\}\), where \(F_n=\sigma(X_1,\dots, X_n)\). The authors obtain sharp bounds for \(E(\max_{k\leq n\leq T}| U_n| ^p)\), \(p\geq 1\), under the condition \(E(f(X_1,\dots,X_k)\mid X_{i_1},\dots,X_{i_h}))\) for all \(\{i_1,\dots,i_h\}\subset\{1,\dots,k\}\), \(h<k\), and, in the case \(1\leq p\leq 2\), under the additional condition of symmetry of \(f\) in its arguments. This is applied to prove the Wald identity \(EU_T=0\), when, for some \(p\in(1,2]\), \(E| f(X_1,\dots,X_k)| ^p<\infty\) and \(ET^\rho<\infty\) with \(\rho=(p^{1/(k-1)}-1)^{-1}\). Several applications are provided.
scientific article; zbMATH DE number 1130495