TITLE: Characteristic function of a non-negative random variable?
QUESTION [2 upvotes]: Is it possible to decide if a random variable is non-negative almost surely, by looking at the characteristic function of the random variable?

REPLY [1 votes]: 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
$$
\lim_{T \to \infty} \frac{1}{2 T} \int_{-T}^{+T} \phi_X(t) dt = 0.
$$
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