The Annals of Probability

Convergence in Distribution of Conditional Expectations

Eimear M. Goggin

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Abstract

Suppose the random variables $(X^N, Y^N)$ on the probability space $(\Omega^N, \mathscr{F}^N, P^N)$ converge in distribution to the pair $(X, Y)$ on $(\Omega, \mathscr{F}, P)$, as $N \rightarrow \infty$. This paper seeks conditions which imply convergence in distribution of the conditional expectations $E^{P^N}\{F(X^N)\mid Y^N\}$ to $E^P\{F(X)\mid Y\}$, for all bounded continuous functions $F$. An absolutely continuous change of probability measure is made from $P^N$ to a measure $Q^N$ under which $X^N$ and $Y^N$ are independent. The Radon-Nikodym derivative $dP^N/dQ^N$ is denoted by $L^N$. Similarly, an absolutely continuous change of measure from $P$ to $Q$ is made, with Radon-Nikodym derivative $dP/dQ = L$. If the $Q^N$-distribution of $(X^N, Y^N, L^N)$ converges weakly to the $Q$-distribution of $(X, Y, L)$, convergence in distribution of $E^{P^N}\{F(X^N)\mid Y^N\}$ (under the original distributions) to $E^P\{F(X)\mid Y\}$ follows. Conditions of a uniform equicontinuity nature on the $L^N$ are presented which imply the required convergence. Finally, an example is given, where convergence of the conditional expectations can be shown quite easily.

Article information

Source
Ann. Probab., Volume 22, Number 2 (1994), 1097-1114.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988743

Digital Object Identifier
doi:10.1214/aop/1176988743

Mathematical Reviews number (MathSciNet)
MR1288145

Zentralblatt MATH identifier
0805.60017

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 94A05: Communication theory [See also 60G35, 90B18]

Keywords
Conditional expectations filtering absolutely continuous change of probability measure Radon-Nikodym derivative equicontinuity

Citation

Goggin, Eimear M. Convergence in Distribution of Conditional Expectations. Ann. Probab. 22 (1994), no. 2, 1097--1114. doi:10.1214/aop/1176988743. https://projecteuclid.org/euclid.aop/1176988743


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