Strong Consistency of Certain Sequential Estimators

Abstract

Motivated by Loynes' (1969) treatment of (weak) consistency of sequential estimators, we establish here some allied results on strong consistency. The strengthened conclusion is achieved by imposing further restrictions, so that our results are not as broadly applicable as Loynes'. The reader is referred to Loynes' paper for additional motivation and discussion. As in that paper, we are concerned with estimators that improve a given one by taking conditional expectations with respect to a sufficient statistic, in the case where the sample size may itself be a random variable. $x_1, x_2, \cdots$ will denote the data sequence, random variables defined on a measurable space $(\Omega, \mathscr{a})$. All probability measures considered on $(\Omega, \mathscr{a})$ will render the sequence iid. It is seen below (Theorem 3.4) that a given sequence $\{t_i\}$ of stopping times for the data sequence leads to a strongly consistent sequence of estimators if $\lim t_i = + \infty$ a.s., and the $t_i$ are $C$-ordered (Definition 2.3). This entails that the $t_i$ increase monotonically to $+ \infty$ a.s., but requires additional structure as well. The specific considerations are in Section 3; Section 2 presents some general notions.