[Math] Which functions of characteristic functions are characteristic functions

characteristic-functionsprobability theoryself-learning

I have a basic question regarding characteristic functions:

If $\phi(t)$ is a characteristic function of some random variable $X$ , then is it necessary for

  1. $\sqrt{\phi(t)}$
  2. $|\phi(t)|$

to be the characteristic function of some other random variable?

I need some help to start .

Best Answer

The function $\varphi\colon t\mapsto\cos t$ is the characteristic function of a random variable taking the values $-1$ and $1$ with probability $1/2$. However, $\left|\varphi\right|$ is not a characteristic function.

Note that since $ \left|\varphi\right|^2$ is a characteristic function (as noted by Qwerty, of the difference of two i.i.d. random variable with characteristic function $\varphi$), this also proves that the square root of a non-negative characteristic function does not need to be a characteristic function.

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