Uniform Integrability in Anscombe’s Theorem for Martingales

Abstract

A useful tool in sequential analysis is Anscombe’s theorem. It asserts that asymptotic normality of standardized sums persists when n is replaced by a random index T_n, provided that T_n/n converges in probability to a finite constant. To obtain moment convergence in Anscombe’s theorem one needs uniform integrability results for standardized random sums, and we investigate this question of uniform integrability in the case of martingale differences.