TITLE: Is $t\mapsto \left|\cos (t)\right|$ a characteristic function?
QUESTION [7 upvotes]: Can anyone explain how I can prove that either $\phi(t) = \left|\cos (t)\right|$ is characteristic function or not? And which random variable has this characteristic function? Thanks in advance.

REPLY [3 votes]: Since $\phi(t) = | \cos(t) |$ is periodic with period $\pi$ and even and if it is valid, it should correspond to a symmetric discrete random variable. 
It is not hard to establish that:
$$
   | \cos(t) | = \frac{2}{\pi} + \frac{4}{\pi} \sum_{m=1}^\infty \frac{(-1)^{m-1}}{4 m^2-1} \cos(2 m t)
$$

Comparing this to $\phi_X(t) = \sum_{m=-\infty}^\infty c_m \mathrm{e}^{i t m}$ we see that $c_4 = - \frac{2}{\pi} \cdot \frac{1}{15}$ is negative, thus can not be a probability of any random variable.