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.
Probability Theory – Is t?|cos(t)| a Characteristic Function?
characteristic-functionsprobabilityprobability distributionsprobability theory
Related Question
- [Math] Knowing that $\phi$ is a characteristic function, show that $e^{\phi-1}$ is also a characteristic function
- Characteristic function for independent $X$ and $Y$
- Can the characteristic function of $\phi(t)$ satisfying $\phi(t)=\phi^2\left(\frac{t}{\sqrt{2}}\right)$ conclude it as C.F. of normal distribution
- Proving a characteristic function is infinitely differentiable
Best Answer
Factoid 1: If a characteristic function is infinitely differentiable at zero, all the moments of the corresponding random variable are finite.
Factoid 2: If all the moments of a random variable are finite, the corresponding characteristic function is infinitely differentiable everywhere on the real line.
Factoid 3: The function $t\mapsto|\cos(t)|$ is infinitely differentiable at $t=0$ but not everywhere on the real line, for example not at $t=\pi/2$.
Ergo.