Signature Cumulants, Ordered Partitions, and Independence of Stochastic Processes

常数扰动下双标度极限 SYK 模型的非交换概率洞察：矩、累积量与 $q$-独立性

Non-commutative probability insights into the double-scaling limit SYK Model with constant perturbations: moments cumulants and $q$-independence

本文研究含固定数目常数对角扰动的 DSSYK 模型，通过矩—累积量关系揭示其矩公式中由 $\ ilde{q}$ 控制的经典与 Boolean 独立性混合，并联系组合结构及引力路径积分。

Bi-monotonic independence for pairs of algebras

In this article, the notion of bi-monotonic independence is introduced as an extension of monotonic independence to the two-faced framework for a family of pairs of algebras in a non-commutative space. The associated cumulants are defined and a moment-cumulant formula is derived in the bi-monotonic setting. In general the bi-monotonic product of states is not a state and the bi-monotonic convolution of probability measures on the plane is not a probability measure. This provides an additional example of how positivity need not be preserved under conditional bi-free convolutions.

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。

通过自由累积量构造乘积 *-概率空间

Construction of product $*$-probability spaces via free cumulants

本文以混合自由累积量消失作为自由性定义，利用自由累积量而非矩来构造 *-概率空间的自由乘积。

自由累积量的分解

Decomposition of free cumulants

引入适应Motzkin路径的非交叉分拆格，定义算子值Motzkin累积量，通过Möbius反演公式细化自由累积量与布尔累积量的关系，并应用于自由概率得到自由累积量的加性分解。

正交不变随机矩阵的无穷小自由性

Infinitesimal freeness for orthogonally invariant random matrices

本文提出实无穷小自由性概念，证明正交不变随机矩阵在特定条件下渐近自由，并引入新的累积量定义及计算公式。

Combinatorial theory of permutation-invariant random matrices II: cumulants, freeness and Levy processes

The $\mathcal{A}$-tracial algebras are algebras endowed with multi-linear forms, compatible with the product, and indexed by partitions. Using the notion of $\mathcal{A}$-cumulants, we define and study the $\mathcal{A}$-freeness property which generalizes the independence and freeness properties, and some invariance properties which model the invariance by conjugation for random matrices. A central limit theorem is given in the setting of $\mathcal{A}$-tracial algebras. A generalization of the normalized moments for random matrices is used to define convergence in $\mathcal{A}$-distribution: t…

分布统计模型：弱矩、累积量及一个中心极限定理

Distributional Statistical Models: Weak Moments, Cumulants, and a Central Limit Theorem

本文提出了一种广义的概率框架，通过将密度替换为由 tempered 分布和 Schwartz 核组成的对，定义了弱矩、弱特征函数和所有阶的弱累积量，从而扩展了经典量并保留了关键的代数性质。主要贡献包括弱累积量的系统代数、弱矩问题的解以及弱中心极限定理的提出，覆盖了经典定理失效的情况。

Categorial Independence and Lévy Processes

We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial Lévy processes on every tensor category with initial unit object and present a construction generalizing the reconstruction of a Lévy process from its …

单调有序非交叉划分上的统计量

Statistics on monotonically ordered non-crossing partitions

本文研究单调有序非交叉划分上的组合统计量，发现期望与方差呈对数增长，并利用树结构简化论证，应用于单调概率的累积量计算。