2014/05/09 by Octavio Arizmend, Takahiro Hasebe, Arizmend, Octavio +1 · 1 citation
Mathematics · #46L54 #60E07 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:46L54 #msc:60E07
paper · pdf · doi:10.48550/arxiv.1405.2162
37 pages
arxiv created 2014/05/09 · arxiv updated 2014/05/12
We study Boolean stable laws, bα,ρ, with stability index α and asymmetry parameter ρ. We show that the classical scale mixture of bα,ρ coincides with a free mixture and also a monotone mixture of bα,ρ. For this purpose we define the multiplicative monotone convolution of probability measures, one is supported on the positive real line and the other is arbitrary. We prove that any scale mixture of bα,ρ is both classically and freely infinitely divisible for α≤1/2 and also for some α>1/2. Furthermore, we show the multiplicative infinite divisibility of bα,1 with respect classical, free and monotone convolutions. Scale mixtures of Boolean stable laws include some generalized beta distributions of 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 Möller. We use a representation of bα,1 as the free multiplicative convolution of a free Bessel law and a free stable law to prove a conjecture of Hinz and Młotkowski regarding the existence of the free Bessel laws as probability measures. The proof depends on the fact that bα,1 has free divisibility indicator 0 for 1/2<α.