A combinatorial method for calculating the moments of Lévy area

Permutations, moments, measures

Which combinatorial sequences correspond to moments of probability measures on the real line? We present a generating function, in the form of a continued fraction, for a fourteen-parameter family of such sequences and interpret these in terms of combinatorial statistics on the symmetric groups. Special cases include several classical and noncommutative probability laws, along with a substantial subset of the orthogonalizing measures in the q-Askey scheme, now given a new combinatorial interpretation in terms of elementary permutation statistics. This framework further captures a variety of in…

Beta矩、组合恒等式与随机游走的统一方法

A Unified Approach to Beta Moments, Combinatorial Identities, and Random Walks

本文提出统一概率方法，将任意维简单对称随机游走的返回概率与矩表示关联，给出涉及Beta和Gamma函数的组合恒等式的概率证明，并推导新恒等式。

Moments, cumulants and diagram formulae for non-linear functionals of random measures

This survey provides a unified discussion of multiple integrals, moments, cumulants and diagram formulae associated with functionals of completely random measures. Our approach is combinatorial, as it is based on the algebraic formalism of partition lattices and Möbius functions. Gaussian and Poisson measures are treated in great detail. We also present several combinatorial interpretations of some recent CLTs involving sequences of random variables belonging to a fixed Wiener chaos.

The optimal order for the p-th moment of sums of independent random variables with respect to symmetric norms and related combinatorial estimates

We calculate the p-the moment of the sum of n independent random variables with respect to symmetric norm in R^n. The order of growth for upper bound p/ln p obtained in ths estimate is optimal. The result extends to generalized Lorentz spaces l_{f,w} under mild assumptions on f. Indeed, the key combinatorial estimate is obtained for the weak l_1 (l_{1,infinity})-norm. Similar results have been obtained independently by Gordon, Litvak, Schuett and Werner for Orlicz norms and by Montgomery-Smith using different techniques and avoiding the combinatorial estimate.

Pseudo-Polynomial Time Algorithm for Computing Moments of Polynomials in Free Semicircular Elements

We consider about calculating $M$th moments of a given polynomial in free independent semicircular elements in free probability theory. By a naive approach, this calculation requires exponential time with respect to $M$. We explicitly give an algorithm for calculating them in polynomial time by rearranging Schützenberger's algorithm.

CUE联合矩问题的一种组合方法

A Combinatorial Approach to the CUE Joint Moment Problem

本文通过组合方法给出了CUE联合矩的精确表达式，统一了相关文献结果，并证明了Basor等人的猜想。