Classical and free infinite divisibility for Boolean stable laws

Abstract

We completely determine the free infinite divisibility for the Boolean stable law which is parametrized by a stability index $\alpha$ and an asymmetry coefficient $\rho$. We prove that the Boolean stable law is freely infinitely divisible if and only if one of the following conditions holds: $0<\alpha\leq\frac{1}{2}$; $\frac{1}{2}<\alpha\leq\frac{2}{3}$ and $2-\frac{1}{\alpha}\leq\rho \leq \frac{1}{\alpha}-1$; $\alpha=1,~\rho=\frac{1}{2}$. Positive Boolean stable laws corresponding to $\rho =1$ and $\alpha \leq \frac{1}{2}$ have completely monotonic densities and they are both freely and classically infinitely divisible. We also show that continuous Boolean convolutions of positive Boolean stable laws with different stability indices are also freely and classically infinitely divisible. Boolean stable laws, free stable laws and continuous Boolean convolutions of positive Boolean stable laws are non-trivial examples whose free divisibility indicators are infinity. We also find that the free multiplicative convolution of Boolean stable laws is again a Boolean stable law.