I would like to prove that for a random variable $X$ and its characteristic function $\phi_X$ the following three properties are equivalent.

$i) \ \phi_X(s) = 1$ for some $s \neq 0$

$ii) \ \phi_X$ is periodic with period $s$

$iii) \ X$ takes values in $\{\frac{2k\pi}{s}: k \in \mathbb{Z}\}$

It is clear that $iii) \implies i)$ since assuming $iii)$ $$\phi_X(t) = \sum_{k \in \mathbb{Z}}exp(it\frac{2\pi k}{s})\mathbb{P}(X=\frac{2\pi k}{s})$$ which immediately gives $\phi_X(s) = \sum_{k\in \mathbb{Z}} \mathbb{P}(X=\frac{2\pi k}{s}) = 1$.

I feel like the remaining implications shouldn't be too difficult but I'm struggling to see where to start with them so I would appreciate any hints.

(I would prefer not to have full solutions since I would still like to think about this problem myself once I know how to start)


I've now managed to solve the question completely myself so I will post my answer here.

$i) \implies iii)$ since assuming $s \neq 0$ and $\phi_X(s) = 1$, $$\mathbb{E}(e^{isX}) = 1$$. Then since $s \in \mathbb{R}$, we have $e^{isX} = 1$ (consider the graph in the argand plane of $e^{isx}$) so X is distributed on the desired set.

Now note that $iii) \implies ii)$ fairly easily since $$\phi_X(t) = \sum_{k \in \mathbb{Z}}exp(it\frac{2\pi k}{s})\mathbb{P}(X=\frac{2\pi k}{s})$$ is clearly periodic with period $s$.

Finally $ii) \implies i)$ since $\phi_X(0) = 1$ and $\phi_X(s) = \phi_X(0)$ by periodicity.

This establishes $i) \iff ii) \iff iii)$

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