## The Annals of Probability

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

### Convergence in Distribution of Conditional Expectations

#### 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