Approximations of distributions of some standardized partial sums in sequential analysis

In sequential analysis it is often necessary to determine the distributions of ?tYt and/or ?a Yt, where t is a stopping time of the form t = inf{n ? 1 : n+Sn+ ?n> a}, Yn is the sample mean of n independent and identically distributed random variables (iidrvs) Yi with mean zero and variance one, Sn is the partial sum of iidrvs Xi with mean zero and a positive finite variance, and {?n} is a sequence of random variables that converges in distribution to a random variable ? as n?? and ?n is independent of (Xn+1, Yn+1), (Xn+2, Yn+2), . . . for all n ? 1. Anscombe's (1952) central limit theorem asserts that both ?t Yt and ?a Yt are asymptotically normal for large a, but a normal approximation is not accurate enough for many applications. Refined approximations are available only for a few special cases of the general setting above and are often very complex. This paper provides some simple Edgeworth approximations that are numerically satisfactory for the problems it considers