I'm working on the following problem and I''m stuck at the begining.
Let $X$ be a lattice distributed random variable, i.e $\exists\, a, b \in \mathbb{R}$ s.t.
$$P[X \in a + b\mathbb{Z}]=1$$
Let $\varphi$ be the characterstic function of $X$.
Show that if $X$ is lattice – distributed, then there exists $u\neq 0$ such that
$$\lvert\varphi(u)\rvert = 1$$
I first noticed that: $\lvert e^{itX}\rvert = 1$
Which could mean that:
$\lvert\varphi(u)\rvert = \lvert e^{itX}\rvert = 1$
Then:
$\varphi(u) = E[e^{iuX}] = e^{itX}$ for some $t\in\mathbb{R}$.
Any help would be appreciated.
Best Answer
If $b=0$ the any $u$ will do. Suppose $b \neq 0$. Take $u=2\pi /b$. We have $\phi (u)=Ee^{(2i\pi/b) X}=\sum e^{2i \pi (\frac a b+n)} P(X=a+b n)=e^{2i \pi (\frac a b)}$ so $|\phi (u)|=1$.