I know the result that says "There is a bijection between distribution functions and characteristic functions".
So I was wondering if there are necessary or sufficient conditions on the characteristic functions so that it is a char. function of a valid random variable. I need conditions that are easy to verify.
For instance, is $\varphi_{X_1,X_2}(x_1,x_2) = (1 + |x_1 x_2| e^{-\frac{(x_1^2 + x_2^2)}2 \sigma^2} )$ a valid characteristic function?
Thank you.