We consider the probability density, often called a generalized Gaussian density, $$p_{\alpha}(t) \propto \exp (- |t|^\alpha),$$ with parameter $0<\alpha<\infty$. For $p = 2$, we recognize a Gaussian distribution and for $p = 1$ a Laplace one. These distributions are known to be infinitely divisible.

For which $\alpha$ is the probability distribution $p_{\alpha}$ infinitely divisible?

Partial answer: The only infinitely divisible distributions that decay faster than $\exp(−O(\lvert t \rvert \log \lvert t \rvert))$ are the Gaussians (see the Theorem 7 of this paper). Hence, for $\alpha>1$ and $\alpha\neq 2$, $p_\alpha$ is not infinitely divisible. I am interested to know what happens in the case $\alpha <1$?