Asymptotic Expansions for the Moments of a Randomly Stopped Average

Abstract

Let $S_1$, $S_2,\cdots$ denote a driftless random walk with values in an inner product space $\mathscr{W}$; let $Z_1$, $Z_2,\cdots$ denote a perturbed random walk of the form $Z_n=n+\langle c,S_n \rangle+\xi_n$, $n = 1, 2,\cdots$, where $\xi_1,\xi_2,\cdots$ are slowly changing, $\langle\centerdot,\centerdot\rangle$ denotes the inner product, and $c\in\mathscr{W}$; and let $t=t_a=inf{n\geq1:Z_n>a}$ for $0\leq a<\infty$. Conditions are developed under which the first four moments of $X_t:=S_t/t$ have asymptotic expansions, and the expansions are found. Stopping times of this form arise naturally in sequential estimation problems, and the main results may be used to find asymptotic expansions for risk functions in such problems. Examples of such applications are included.