conditional expectations are uniformly integrable

The collection of all conditional expectations of an integrable random variable forms a uniformly integrable set. More generally, we have the following result.

Theorem.

Let $S$ be a uniformly integrable set of random variables defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ . Then, the set

$\left\{\mathbb{E}[X\mid\mathcal{G}]:\textrm{$X\in S$ and $\mathcal{G}$ is a % sub-$\sigma$-algebra of $\mathcal{F}$}\right\}$

is also uniformly integrable.

To prove the result, we first use the fact that uniform integrability implies that $S$ is $L^{1}$ -bounded. That is, there is a constant $L>0$ such that $\mathbb{E}[|X|]\leq L$ for every $X\in S$ . Also, choosing any $\epsilon>0$ , there is a $\delta>0$ so that

$\mathbb{E}[|X|1_{A}]<\epsilon$

for all $X\in S$ and $A\in\mathcal{F}$ with $\mathbb{P}(A)\leq\delta$ .

Set $K=L/\delta$ . Then, if $Y=\mathbb{E}[X\mid\mathcal{G}]$ for any $X\in S$ and $\mathcal{G}\subseteq\mathcal{F}$ , Jensen’s inequality gives

$|Y|\leq\mathbb{E}[|X|\mid\mathcal{G}].$

So, applying Markov’s inequality,

$\mathbb{P}(|Y|>K)\leq K^{-1}\mathbb{E}[|Y|]\leq K^{-1}\mathbb{E}[|X|]\leq L/K=\delta$

and, therefore

$\mathbb{E}[|Y|1_{\{|Y|>K\}}]\leq\mathbb{E}[|X|1_{\{|Y|>K\}}]<\epsilon.$

Title	conditional expectations are uniformly integrable
Canonical name	ConditionalExpectationsAreUniformlyIntegrable
Date of creation	2013-03-22 18:40:08
Last modified on	2013-03-22 18:40:08
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	5
Author	gel (22282)
Entry type	Theorem
Classification	msc 28A20
Classification	msc 60A10
Related topic	ConditionalExpectation
Related topic	UniformlyIntegrable