ASYMPTOTIC EXPANSIONS FOR THE MOMENTS OF A RANDOMLY STOPPED AVERAGE: EXTENSION AND APPLICATIONS OF A RESULT OF ARAS AND WOODROOFE

Aras and Woodroofe (1993) provide asymptotic expansions of the first four moments of ¯ Xt := St/t where t = ta =i nf{n ≥ 1: Zn >a }, Zn = n + YT. Martinsek (1983) obtained the second order expansion of the risk of this sequential estimation procedure, assuming the initial sample size m →∞ at a certain rate (but without specifying the form of distribution). If the initial sample size m is assumed to be prefixed, the second order expansion of the risk has been established by Woodroofe (1977) but only for normally distributed Yi .T he present paper provides the second order expansion of the risk under assumptions that m is prefixed and that the Yi is continuous with a bounded probability density function.