Checking whether they are characteristic functions or not.

Question

My problem is determined whether the following functions $\phi(t)=\cos(t)$ and $\phi(t)=(1+t^4)^{-1}$ are characteristic functions for some random variable?

One thing I know that if they didn't satisfy properties of characteristic functions (such as $\phi(0)=1$, and $\phi$ is uniformly continuous), then we can say they are not characteristic functions. However, these two functions agrees these properties, and hence I may guess they will be characteristic functions for some random variables. However, I have no idea how to show the argument.

Thank you.

Answer 1

$$\cos(t) = \frac{e^{it} + e^{-it}}{2}$$

so this is the characteristic function of a random variable that is $+1$ with probability $1/2$ and $-1$ with probability $1/2$.

$(1+t^4)^{-1}$ is the Fourier transform of $2^{-3/2} e^{-|s|/\sqrt{2}} (\cos(s/\sqrt{2}) + \sin(|s|/\sqrt{2})$ but this is negative e.g. for $s/\sqrt{2} \in (3\pi/4, 7\pi/4)$, so in this case $\phi$ can't be a characteristic function.

EDIT: Another way to show $\phi(t) = (1+t^4)^{-1}$ is not a characteristic function is to show that there is an $n$-tuple of real numbers $s_1, \ldots, s_n$ such that the $n \times n$ matrix with entries $A_{jk} = \phi(s_j - s_k)$ is not positive semidefinite. For example, this is the case with $n=3$ and $s_1 = 1/2,s_2 = 1, s_3 = 3/2)$: you can check that $$ \det\pmatrix{1 & 16/17 & 1/2\cr 16/17 & 1 & 16/17\cr 1/2 & 16/17 & 1} < 0$$