Moments of randomly stopped $U$-statistics
Copyright policy )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
- King’s University United States
- Stanford University United States
Stopping times; optimal stopping problems; gambling theory, Wald identity, $U$-statistics, Decoupling inequalities, stopping times, \(L^p\)-limit theorems, Sequential estimation, martingales, 62L12, 60F25, Wald’s equation, uniform integrability, 60G40
Stopping times; optimal stopping problems; gambling theory, Wald identity, $U$-statistics, Decoupling inequalities, stopping times, \(L^p\)-limit theorems, Sequential estimation, martingales, 62L12, 60F25, Wald’s equation, uniform integrability, 60G40
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