Almost Sure Equiconvergence of Conditional Expectations

Abstract

If $(X, \mathscr{F}, P)$ is a probability space then a pseudo-metric $\delta$ can be defined on the sub-$\sigma$-fields of $\mathscr{F}$ by $\delta(\mathscr{A, B}) = \sup_{A \in \mathscr{A}}\inf_{B \in \mathscr{B}}P(A \Delta B) \vee \sup_{B \in \mathscr{B}}\inf_{A \in \mathscr{A}}P(A \Delta B).$ Boylan, Neveu, and Rogge, among others, have considered equiconvergence of conditional expectations of uniformly bounded measurable functions given sub-$\sigma$-fields $\{\mathscr{F}_n:1 \leq n \leq \infty\}$ in probability and in $L_p, 1 \leq p < \infty$, as $\delta(\mathscr{F}_n, \mathscr{F}_\infty) \rightarrow 0$. This paper proves the corresponding almost sure equiconvergence results when $\mathscr{F}_n \uparrow \mathscr{F}_\infty$ or $\mathscr{F}_n \downarrow \mathscr{F}_\infty$. A sharp uniform bound for the rate of convergence is given. A consequence is that if $\mathscr{F}_n \uparrow \mathscr{F}_\infty$ or $\mathscr{F}_n \downarrow \mathscr{F}_\infty$ then the sequence of conditional expectations given $\mathscr{F}_n$ converges uniformly for all uniformly bounded measurable functions to the conditional expectation given $\mathscr{F}_\infty$ if and only if $\delta(\mathscr{F}_n, \mathscr{F}_\infty) \rightarrow 0$.