自由概率论中由 Magnus 展开得到的累积量-累积量关系

Stein approximation for multidimensional Poisson random measures by third cumulant expansions

We obtain Stein approximation bounds for stochastic integrals with respect to a Poisson random measure over ${\Bbb R}^d$, $d\geq 2$. This approach relies on third cumulant Edgeworth-type expansions based on derivation operators defined by the Malliavin calculus for Poisson random measures. The use of third cumulants can exhibit faster convergence rates than the standard Berry-Esseen rate for some sequences of Poisson stochastic integrals.

非线性滤波中小系统噪声展开的有限累积量塔

Finite Cumulant Towers for Small-System-Noise Expansions in Nonlinear Filtering

该研究针对小系统噪声下的非线性滤波，提出有限累积量塔方法，可递归求解条件累积量展开系数，减少常微分方程变量数，优于现有算法。

Forests, cumulants, martingales

This work is concerned with forest and cumulant type expansions of general random variables on a filtered probability spaces. We establish a "broken exponential martingale" expansion that generalizes and unifies the exponentiation result of Al{ò}s, Gatheral, and Radoičić and the cumulant recursion formula of Lacoin, Rhodes, and Vargas. Specifically, we exhibit the two previous results as lower dimensional projections of the same generalized forest expansion, subsequently related by forest reordering. Our approach also leads to sharp integrability conditions for validity of the cumulant formula…

有限自由累积量：乘积卷积、亏格展开与无穷小分布

Finite Free Cumulants: Multiplicative Convolutions, Genus Expansion and Infinitesimal Distributions

本文给出了多项式乘积卷积的有限自由累积量的组合公式及其拓扑展开，揭示了有限自由概率与自由概率的联系，并利用该展开研究了多项式的无穷小分布，最后为多项式导数根分布与自由分数卷积幂的联系提供了新证明。

Jack polynomials and free cumulants

We study the coefficients in the expansion of Jack polynomials in terms of power sums. We express them as polynomials in the free cumulants of the transition measure of an anisotropic Young diagram. We conjecture that such polynomials have nonnegative integer coefficients. This extends recent results about normalized characters of the symmetric group.

Genus expansion for real Wishart matrices

We present an exact formula for moments and cumulants of several real compound Wishart matrices in terms of an Euler characteristic expansion, similar to the genus expansion for complex random matrices. We consider their asymptotic values in the large matrix limit: as in a genus expansion, the terms which survive in the large matrix limit are those with the greatest Euler characteristic, that is, either spheres or collections of spheres. This topological construction motivates an algebraic expression for the moments and cumulants in terms of the symmetric group. We examine the combinatorial pr…

Analytical probabilistic approach to the ground state of lattice quantum systems: exact results in terms of a cumulant expansion

We present a large deviation analysis of a recently proposed probabilistic approach to the study of the ground-state properties of lattice quantum systems. The ground-state energy, as well as the correlation functions in the ground state, are exactly determined as a series expansion in the cumulants of the multiplicities of the potential and hopping energies assumed by the system during its long-time evolution. Once these cumulants are known, even at a finite order, our approach provides the ground state analytically as a function of the Hamiltonian parameters. A scenario of possible applicati…

经典集合的光滑多迹统计：大N展开、累积量和矩阵积分

Smooth Multi-Trace Statistics of Classical Ensembles: Large $N$ Expansions, Cumulants, and Matrix Integrals

本文研究了经典集合中光滑多迹统计的大 $N$ 展开，推导了累积量的渐近消失、中心极限定理以及矩阵积分自由能和可观测量的渐近展开。

基于累积量的Edgeworth展开的α-混合随机场的Wasserstein-p界

Wasserstein-p Bounds via Cumulant-Based Edgeworth Expansion for $α$-Mixing Random Fields

本文将Wasserstein-p界扩展至α-混合随机场，通过累积量Edgeworth展开与Stein方法，证明其重标平均值以O(|T|^(-β))收敛于标准正态分布，β可达1/2。

Real and quaternionic second-order free cumulants and connections to matrix cumulants

We present definitions for real and quaternionic second-order free cumulants, functions whose collective vanshing when applied to elements from different subalgebras is equivalent to the second-order real (resp.\ quaternionic) freeness of those subalgebras. We construct a poset related to the annular noncrossing partitions, and calculate the Möbius function, which may be used to compute the coefficients in the expression for the second-order free cumulants. We show the connection between second-order free cumulants and the topological expansion interpretation of matrix cumulants. This provides…