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August, 1984 Almost Sure Equiconvergence of Conditional Expectations
H. G. Mukerjee
Ann. Probab. 12(3): 733-741 (August, 1984). DOI: 10.1214/aop/1176993224


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 $\delta(\mathscr{F}_n, \mathscr{F}_\infty) \rightarrow 0$.


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H. G. Mukerjee. "Almost Sure Equiconvergence of Conditional Expectations." Ann. Probab. 12 (3) 733 - 741, August, 1984.


Published: August, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0557.28001
MathSciNet: MR744230
Digital Object Identifier: 10.1214/aop/1176993224

Primary: 28A20
Secondary: 60645

Keywords: a.s. equiconvergence , conditional expectation , metric for $\sigma$-fields

Rights: Copyright © 1984 Institute of Mathematical Statistics

Vol.12 • No. 3 • August, 1984
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