The Annals of Probability

Almost Sure Equiconvergence of Conditional Expectations

H. G. Mukerjee

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

Article information

Source
Ann. Probab., Volume 12, Number 3 (1984), 733-741.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993224

Digital Object Identifier
doi:10.1214/aop/1176993224

Mathematical Reviews number (MathSciNet)
MR744230

Zentralblatt MATH identifier
0557.28001

JSTOR
links.jstor.org

Subjects
Primary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
Secondary: 60645

Keywords
Conditional expectation a.s. equiconvergence metric for $\sigma$-fields

Citation

Mukerjee, H. G. Almost Sure Equiconvergence of Conditional Expectations. Ann. Probab. 12 (1984), no. 3, 733--741. doi:10.1214/aop/1176993224. https://projecteuclid.org/euclid.aop/1176993224


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