CLASSICAL SCALE MIXTURES OF BOOLEAN STABLE LAWS

We study Boolean stable laws, b(alpha,rho), with stability index b(alpha,rho) and asymmetry parameter rho. We show that the classical scale mixture of b(alpha,rho) coincides with a free mixture and also a monotone mixture of b(alpha,rho). For this purpose we define the multiplicative monotone convolution of probability measures, one supported on the positive real line and the other arbitrary. We prove that any scale mixture of b(alpha,rho) is both classically and freely infinitely divisible for alpha <= 1/2 and also for some alpha > 1/2. Furthermore, we show the multiplicative infinite divisibility of b(alpha), 1 with respect to classical, free and monotone convolutions. Scale mixtures of Boolean stable laws include some generalized beta distributions of the second kind, which turn out to be both classically and freely infinitely divisible. One of them appears as a limit distribution in multiplicative free laws of large numbers studied by Tucci, Haagerup and Moller. We use a representation of b(alpha), 1 as the free multiplicative convolution of a free Bessel law and a free stable law to prove a conjecture of Hinz and Mlotkowski regarding the existence of the free Bessel laws as probability measures. The proof depends on the fact that b(alpha), 1 has free divisibility indicator 0 for 1/2 < alpha.