Positivity of exponentially bounded characteristic functions

Question

I've noticed that for the classical examples of exponentially bounded, symmetrical distributions (Gaussian, Laplace, Double Exponential, Uniform), their characteristic functions are positive for all frequencies. This doesn't seem to be the case for many symmetrical distributions with fat tails (like the double Gamma distribution).

Is there a theorem that states that characteristic functions of exponentially bounded, symmetrical distributions in $\mathbb{R}$ are positive? Is there a counterexample I'm missing?

Answer 1

The rate of decrease of the tails of a probability distribution has to do with the degree of smoothness of the corresponding characteristic function (c.f.).

The rate of decrease of the tails has nothing to do with the positivity of the c.f. E.g., the uniform distribution over the interval $[-1,1]$ has zero tails, but its c.f. $f$, given by $f(t)=\dfrac{\sin t}t$ for $t\ne0$, is not everywhere positive.