Free Regular Infinite Divisibility and Squares of Random Variables with $\boxplus$-infinitely Divisible Distributions

Details Free Regular Infinite Divisibility and Squares of Random Variables with $\boxplus$-infinitely Divisible Distributions Free Regular Infinite Divisibility and Squares of Random Variables with $\boxplus$-infinitely Divisible Distributions

2011-12-31

arxiv.org/abs/1201.0311v1

Mathematics

Probability

16 pages

Scientific paper

We study the freely infinitely divisible distributions corresponding to the classically infinitely divisible distributions with positive support: the free regular infinitely divisible measures. We clarify the role of these measures as the distributions of free subordinators. We show that the square of a symmetric freely infinitely divisible distribution is also freely infinitely divisible. Moreover, it can be represented as the free multiplicative convolution of a free Poisson and a free regular measure. This gives two new explicit examples of measures which are infinitely divisible with respect to both classical and free convolutions: \chi^2 and F(1,1). Another consequence is that the free commutator preserves free infinite divisibility. We also prove that the class of free regular measures is closed under the free multiplicative convolution, t-th boolean power for 01 and weak convergence.

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