Cumulants in noncommutative probability I. Noncommutative Exchangeability Systems

Cumulants in Noncommutative Probability Theory IV. De Finetti's Theorem and $L^p$-Inequalities

In this paper we collect a few results about exchangeability systems in which crossing cumulants vanish, which we call noncrossing exchangeability systems. The main result is a free version of De Finetti's theorem, characterising amalgamated free products as noncrossing exchangeability systems which satisfy a so-called weak singleton condition. The main tool in the proof is an $L^p$-inequality with uniformly bounded constants for i.i.d. sequences in noncrossing exchangeability systems.

Cumulants in noncommutative probability II. Generalized Gaussian random variables

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

Relations between cumulants in noncommutative probability

We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed.

Cumulants in Noncommutative Probability Theory III. Creation- and annihilation operators on Fock spaces

Fock space constructions give rise to natural exchangeable families and are thus well suited for cumulant calculations. In this paper we develop some general formulas and compute cumulants for generalized Toeplitz operators, notably for q-Fock spaces, previously considered by M. Anshelevich and A. Nica, and Fock spaces for characters of the infinite symmetric group, which where constructed by Bozejko and Guta. An expression for cumulants in terms of the cycle-cover polynomials of certain directed graphs is obtained in this case.

累积量、可传播性与 Campbell-Baker-Hausdorff 级数

Cumulants, Spreadability and the Campbell-Baker-Hausdorff Series

本文提出可传播性系统以推广可交换性系统，统一非交换概率中的独立性与累积量概念，涵盖单调独立性等情形，并基于关联代数与 Faa di Bruno 定理证明其涵盖已知累积量，同时揭示了 Campbell-Baker-Hausdorff 级数与 Goldberg 系数的新推导。

Free cumulants, Schröder trees, and operads

The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Faà di Bruno algebra, and then to the group of a free operad over Schröder trees. This leads to new combinatorial expressions, which remain valid for operator-valued free probability. Specializations of these expressions give back Speicher's formula in terms of noncrossing partitions, and its interpretation in terms of characters due to Ebrahimi-Fard and Patras.