Franz Lehner

Cumulants in noncommutative probability theory I. Noncommutative exchangeability systems

Abstract.Cumulants linearize convolution of measures. We use a formula of Good to define noncommutative cumulants in a very general setting. It turns out that the essential property needed is exchangeability of random variables. Roughly speaking the formula says that cumulants are moments of a certain ‘‘discrete Fourier transform’’ of a random variable. This provides a simple unified method to understand the known examples of cumulants, like classical, free and various q-cumulants.

Cumulants in noncommutative probability theory II

We continue the investigation of noncommutative cumulants. In this paper various characterizations of noncommutative Gaussian random variables are proved.

The normal distribution is -infinitely divisible

Abstract We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a sub-family of Askey–Wimp–Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case. At the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution, the others being the Cauchy distribution and the free 1/2-stable distribution.

Free Cumulants and Enumeration of Connected Partitions

Abstract A combinatorial formula is derived which expresses free cumulants in terms of classical cumulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connected partitions, respectively. The proof relies on Mobius inversion on the partition lattice.

On the spectrum of lamplighter groups and percolation clusters

Let

Relations between cumulants in noncommutative probability

{\mathfrak{G}}

Cumulants, lattice paths, and orthogonal polynomials

be a finitely generated group and X its Cayley graph with respect to a finite, symmetric generating set S. Furthermore, let