Given characteristic function $\phi(t) = e^{-|t|^{a}/a}$, where a is some constant

Question

I was wondering about the following question:

I am given a family of distributions whose characteristic functions are of the form $\phi(t) = e^{-|t|^{a}/a}$, where a is some constant. We want to find out what values can a have such that a distribution can exist.

Notice that

• In the case of a = 2, $e^{-|t|^{2}/2}$ is the characteristic function of a standard normal.

• In the case of a = 1, $e^{-|t|}$ is the characteristic function of a standard Cauchy.

What I noticed is that using inversion formula, we know the characteristic function $\phi(t)$ must be such that $\int_{\mathbb{R}} |\phi(t)| dt < \infty$. This is not true here for a $\leq 0$.

However, I wonder if the inversion formula tells me more than this about what values a can take, or is it just a formula telling me how to compute the density function from a characteristic function so long as the preconditions are met (which is true for a $>0$ I believe).

Any comment / ideas / discussions are welcome. Thanks a lot!

Answer 1

These characteristic functions are well known. $\phi (t)$ is a characteristic function iff $0<a \leq 2$. For $a=2$ we get the normal distribution and for $0<a<2$ we get a so-called (symmetric) stable distribution. $a=1$ gives a multiple of a Cauchy random variable. You can search Wikipedia for a discussion of stable distributions.