Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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 $X$ be a real random variable and $\phi$ its characteristic function. Show that there exist real $a,b$ such that $P\left(X \in a+b\mathbb{Z} \right) = 1$ if and only if there exists a nonzero $x$ such that $\left|\phi(x)\right| = 1$.

Any ideas or hints would be appreciated. Thanks.

share|cite|improve this question
Hint: How can the weighted average of points on a unit circle not end up in the interior of the circle? – Thomas Andrews Dec 4 '12 at 17:09
Any attempt would be appreciated. – Davide Giraudo Dec 4 '12 at 17:14

You can start with this : according to the triangle inequality, $|\phi(\theta)| \leq E(|e^{i\theta X}|)=1$ with equality if and only if there exists $u \in \mathbb{C}$ with $|u|=1$ such that $e^{i\theta X} = u$ a.e.

share|cite|improve this answer
+1 for the carefully dosed indication. – Did Dec 4 '12 at 18:47

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.