Embedding submartingales in wiener processes with drift, with applications to sequential analysis

Abstract

SummarySkorokhod (1961) demonstrated how the study of martingale sequences (and zero-mean random walks) can be reduced to the study of the Wiener process (without drift) at a sequence of random stopping times. We show how the study of certain submartingale sequences, including certain random walks with drift and log likelihood ratio sequences, can be reduced to the study of the Wiener process with drift at a sequence of stopping times (Theorem 4.1). Applications to absorption problems are given. Specifically, we present new derivations of a number of the basic approximations and inequalities of classical sequential analysis, and some variations on them — including an improvement on Wald's lower bound for the expected sample size function (Corollary 7.5).