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 a multiple of k.First, we consider the combinatorial convolution * in the lattices N C of noncrossing partitions and N C k of k-divisible non-crossing partitions and show that convolving k times with the zeta-function in N C is equivalent to convolving once with the zeta-function in N C k . Furthermore, when x is k-divisible, we derive a formula for the free cumulants of x k 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 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 → µ k . 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.

Non-commutative Probability Theory for Topological Data Analysis

“…2 in the sense zz * = z * z. (4). A (normal) random variable u ∈ A is a Haar-distributed unitary random variable if it is unitary (i.e.…”

“…Since ∆ t is a simplicial complex, we may calculate topological invariants with the computer, and visualize the evolution of the Betti numbers (or other invariants) as t grows (see Figure 4) 4 .…”

“…Interval and non-crossing partitions have nice arithmetic features. In particular, the sub-posets of k-divisible partitions may be considered and their Möbius theory can be calculated (see [4]). These have been used in order to compute moments of products of free random variables [8].…”