Take the 2-minute tour ×  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.  share|improve this question You might find this
interesting –  Sasha Sep 29 '11 at 5:16  1 Answer 1  up vote 4
down vote accepted  Bochner's theorem characterizes if a
characteristic function corresponds to the random variable, but
it's not an easy criterion.  It says that for all $x_1, \ldots,
x_m \in \mathbb{R}^2$ and $c_1,\ldots,c_m \in \mathbb{C}$ the
characteristic function $\phi$ should verify $$ \sum_{i=1}^m
\sum_{j=1}^m \bar{c}_i c_j \phi(x_i - x_j) \ge 0 $$  For the
characteristic function in question, choose $m=2$, and