Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let X and Y be bounded random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider two independent sub-$\sigma$ fields $\mathcal{G}$ and $\mathcal{H}$ of $\mathcal{F}$. We assume X is $\mathcal{G}$-measurable and Y is $\mathcal{H}$-measurable. Prove the following equality holds: $$\mathbb{E} (XY\,\, |\, \sigma (\mathcal{H},\mathcal{G})) = \mathbb{E} (X\,\, |\mathcal{H}) \mathbb{E} (Y\,\, |\mathcal{G})$$

share|improve this question

1 Answer 1

up vote 3 down vote accepted

It doesn't.

If $X$ is $\mathcal G$-measurable and $Y$ is $\mathcal H$-measurable then $XY$ is $\sigma(\mathcal G, \mathcal H)$-measurable. Therefore $\mathbb E(XY|\sigma(\mathcal G, \mathcal H))=XY$.

But if $\mathcal G$ and $\mathcal H$ are independent then $\mathbb E(X|\mathcal H)= \mathbb E(X)$ and $\mathbb E(Y|\mathcal G)= \mathbb E(Y)$.

So you claim that $XY = \mathbb E(X)\mathbb E(Y)$, which holds only if $X$ and $Y$ are almost surely constant.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.