Wald's Equation for a Class of Denormalized $U$-Statistics

Abstract

Under suitable conditions on a stopping time $T$ and zero mean i.i.d. random variables $\{X_n, n \geq 1\}$, a Wald-type equation $ES_{k, T} = 0$ is obtained where $S_{k, n}$ is the sum of products of $k$ of the $X$'s with indices from 1 to $n$. This, in turn, is utilized to obtain information about the moments of $T_k = \inf\{n \geq k: S_{k, n} \geq 0\}$ and $W_c = \inf\{n \geq 2: S^2_{1, n} > c\sum^n_{j = 1}X^2_j\}, c > 0$.