期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2016
卷号：XIII
页码：309-336
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：We prove that Xr follows a free regular distribution, i.e. the law of anonnegative free L´evy process if: (1) X follows a free Poisson distribution withoutan atom at 0 and r ∈ (−∞, 0] ∪ [1,∞); (2) X follows a free Poisson distributionwith an atom at 0 and r ≥ 1; (3) X follows a mixture of some HCM distributionsand |r| ≥ 1; (4) X follows some beta distributions and r is taken from someinterval. In particular, if S is a standard semicircular element then |S|r is freelyinfinitely divisible for r ∈ (−∞, 0]∪[2,∞). Also we consider the symmetrization ofthe above probability measures, and in particular show that |S|r sign(S) is freelyinfinitely divisible for r ≥ 2. Therefore Sn is freely infinitely divisible for everyn ∈ N. The results on free Poisson and semicircular random variables have a goodcorrespondence with classical ID properties of powers of gamma and normal randomvariables.
