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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.

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I'd be happy to help, if you do something about that awful accept rate. –  Stefan Hansen Dec 4 '12 at 17:00
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
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1 Answer

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.

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+1 for the carefully dosed indication. –  Did Dec 4 '12 at 18:47
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