Characteristic function of a non-negative random variable?

Question

Is it possible to decide if a random variable is non-negative almost surely, by looking at the characteristic function of the random variable?

Answer 1

Partial answer: According to a theorem (Gil-Pelaez), if $0$ is a point of continuity of $F_X$, then $$ F_X(0) = \frac{1}{2} - \frac{1}{\pi} \int_0^{\infty} \frac{\mathrm{Im}[\phi_X(t)]}{t} dt, $$ so, since a random variable $X$ is a.s. non-negative, then $F_X(0) = 0$, $\int_0^{\infty} \frac{\mathrm{Im}[\phi_X(t)]}{t} dt = \pi / 2$ is the condition for a.s. non-negativity in that case.

$F_X$'s continuity at $0$ can be tested by checking if $$ \lim_{T \to \infty} \frac{1}{2 T} \int_{-T}^{+T} \phi_X(t) dt = 0. $$