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.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let X and Y be two random variables on some $(\Omega, \mathcal{F}, \mathbb{P}).$ We can assume that Y is bounded. It is clear that X and Y are independent iff:

$$\mathbb{E}[e^{iuX}e^{ivY}]=\mathbb{E}[e^{iuX}]\;\mathbb{E}[e^{ivY}], \ \forall u,v \in \mathbb{R}.$$

My question is, can we show that X and Y are independent iff

$$\mathbb{E}[e^{iuX}p(Y)]=\mathbb{E}[e^{iuX}]\;\mathbb{E}[p(Y)]$$ $\forall u \in \mathbb{R} $ and for all polynomial functions $p(\cdot)$ ?

Thank you!

share|cite|improve this question
up vote 2 down vote accepted

Yes. If $|Y| \le B$, we can approximate $\exp(ivy)$ uniformly by polynomials $p(y)$ on $[-B,B]$, and since $|e^{iuX}| = 1$ this approximates ${\mathbb E}[e^{iuX} e^{ivY}]$ and ${\mathbb E}[e^{ivY}]$ by ${\mathbb E}[e^{iuX} p(Y)]$ and ${\mathbb E}[p(Y)]$ respectively.

share|cite|improve this answer
thanks for the answer! – user7762 Sep 6 '12 at 23:59

Your Answer


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.