K-divisible random variables in free probability

Abstract

We introduce and study the notion of k-divisible elements in a non-commutative probability space. A k-divisible element is a (non-commutative) random variable whose n-th moment vanishes whenever n is not divisible by k.First, we consider the combinatorial convolution ⁎ in the lattices NC of non-crossing partitions and NCk of k-divisible non-crossing partitions and show that convolving k times with the zeta function in NC is equivalent to convolving once with the zeta function in NCk. Furthermore, when x is k-divisible, we derive a formula for the free cumulants of xk in terms of the free cumulants of x, involving k-divisible non-crossing partitions.Second, we prove that if a and s are free and s is k-divisible then sps and a are free, where p is any polynomial (on a and s) of degree k−2 on s. Moreover, we define a notion of R-diagonal k-tuples and prove similar results.Next, we show that free multiplicative convolution between a measure concentrated in the positive real line and a probability measure with k-symmetry is well defined. Analytic tools to calculate this convolution are developed.Finally, we concentrate on free additive powers of k-symmetric distributions and prove that μ⊞t is a well defined k-symmetric probability measure, for all t>1. We derive central limit theorems and Poisson type ones. More generally, we consider freely infinitely divisible measures and prove that free infinite divisibility is maintained under the mapping μ→μk. We conclude by focusing on (k-symmetric) free stable distributions, for which we prove a reproducing property generalizing the ones known for one sided and real symmetric free stable laws.