Is there a bounded random variable whose characteristic function decays exponentially in the tails?
The c.f. of uniform distribution $U(-1,1)$ is $\frac{\sin t}{t}$. I can concolute more $U(-1,1)$ together to get $\frac{\sin^{m}t}{t^{m}}$ to make the tail of it more thinner.
No matter how many terms I convolve, the decay of the characteristic function tails is only at a polynomial rate. I want to know if there exists a bounded random variable whose characteristic function decays exponentially in the tails, like the characteristic function of a normal distribution $e^{-\frac{t^{2}}{2}}$ or a Cauchy distribution $e^{-|t|}$.