The linearization of the central limit operator in free probability theory

On Random Operator-Valued Matrices: Operator-Valued Semicircular Mixtures and Central Limit Theorem

Motivated by a random matrix theory model from wireless communications, we define random operator-valued matrices as the elements of $L^{\infty-}(Ω,{\mathcal F},{\mathbb P}) \otimes M_d({\mathcal A})$ where $(Ω,{\mathcal F},{\mathbb P})$ is a classical probability space and $({\mathcal A},φ)$ is a non-commutative probability space. A central limit theorem for the mean $M_d(\mathbb{C})$-valued moments of these random operator-valued matrices is derived. Also a numerical algorithm to compute the mean $M_d({\mathbb C})$-valued Cauchy transform of operator-valued semicircular mixtures is analyzed.

Matricial R-transform

We study the addditon problem for strongly matricially free random variables which generalize free random variables. Using operators of Toeplitz type, we derive a linearization formula for the `matricial R-transform' related to the associated convolution. It is a linear combination of Voiculescu's R-transforms in free probability with coefficients given by internal units of the considered array of subalgebras. This allows us to view this formula as the `matricial linearization property' of the R-transform. Since strong matricial freeness unifies the main types of noncommutative independence, t…

Free Probability for Pairs of Faces I

We consider a notion of bi-freeness for systems of non-commutative random variables with two faces, one of left variables and another of right variables. This includes bi-free convolution operations, bi-free cumulants and the bi-free central limit.

A quadratic regression problem for two-state algebras with application to the Central Limit Theorem

We extend a free version of the Laha-Lukacs theorem to probability spaces with two-states. We then use this result to generalize a noncommutative CLT of Kargin to the two-state setting.

有限自由概率的奇异值理论

A theory of singular values for finite free probability

本文提出适用于矩形矩阵的有限自由概率理论，将其转化为多项式奇异值操作，证明其能复制自由概率变换并渐近收敛，同时在新框架下显式化大数定律与中心极限定理。

矩阵集中不等式与自由概率

Matrix Concentration Inequalities and Free Probability

本文发展了能捕捉非交换性的高斯随机矩阵谱非渐近界，量化了其与自由概率论非交换模型的谱的关系，在经典不等式次优的情形下获得了紧的结果，并推广至一般独立随机矩阵和。

张量自由概率理论：渐近张量自由性与中心极限定理

Tensor free probability theory: asymptotic tensor freeness and central limit theorem

本研究推广了Voiculescu的渐近自由独立性，针对局部幺正不变的张量积结构随机矩阵提出了张量自由性概念，通过混合张量自由累积量的消失来定义联合分布，并建立了张量自由版本的中心极限定理。

Multilinear function series in conditionally free probability with amalgamation

As in the cases of freeness and monotonic independence, the notion of conditional freeness is meaningful when complex-valued states are replaced by positive conditional expectations. In this framework, the paper presents several positivity results, a version of the central limit theorem and an analogue of the conditionally free R-transform constructed by means of multilinear function series.

Stable actions and central extensions

A probability-measure-preserving action of a countable group is called stable if its transformation-groupoid absorbs the ergodic hyperfinite equivalence relation of type II_1 under direct product. We show that for a countable group G and its central subgroup C, if G/C has a stable action, then so does G. Combining a previous result of the author, we obtain a characterization of a central extension having a stable action.