arXiv subjects

Pei-Lun Tseng

Publications and source records attributed to Pei-Lun Tseng.

9 recordsLinked to original sources

arXiv2026

Asymptotic infinitesimal freeness of covariance matrices

We consider $n\times n$ covariance matrices $M=\frac{1}{n}XX^*$ where $X=(x_{i,j})$ is a matrix whose entries are independent complex random variables with $\mathbb{E}(x_{i,j})=0$ and $\mathbb{E}(|x_{i,j}|^2)=1$. We derive a $\frac{1}{n}$ expansion of the mixed moments, $\frac{1}{n}\mathbb{E}(\Tr(M^{(r_1)}\cdots M^{(r_q)}))$, of the form $a_0+a_1\frac{1}{n}+O(\frac{1}{n^2})$. This permits us to find explicit formulas for the moments and infinitesimal moments of several covariance matrices where we allow repetition. As an application of our formulas, we derive asymptotic freeness and infinitesimal freeness of independent covariance matrices under a fourth-moment condition. This generalizes previous results for the Wishart ensemble in which $x_{i,j}$ is complex Gaussian.

math.PR↗

arXiv2026

Scaling limit theorem for mixed free and Boolean convolution powers

We prove a scaling limit theorem for a double sequence of probability measures involving additive free convolution $\boxplus$ and additive Boolean convolution $\uplus$. Let $μ$ be a probability measure on $\mathbb{R}$ with mean zero and variance one, and let $M=M(N)>0$ satisfy $MN^{α+1/2}\to t>0$. We study the weak limits, as $N\to \infty$, of the double arrays $D_{N^α}((μ^{\boxplus N})^{\uplus M})$. We show that the limit distribution is the Cauchy distribution with scale parameter $t$ if $α>-1/2$, the $t$-fold Boolean convolution power of the standard semicircle law if $α=-1/2$, and the point mass at the origin if $α<-1/2$.

math.PR↗

arXiv2025

Rates of convergence in the Free Multiplicative Central Limit Theorem

We provide the first quantitative estimates for the rate of convergence in the free multiplicative central limit theorem (CLT), in terms of the Kolmogorov and $r$-Wasserstein distances for $r \geq 1$. While the free additive CLT has been thoroughly studied, including convergence rates, the multiplicative setting remained open in this regard. We consider products of the form $$ π_n^{g,n^{-1/2}x} := g\left(\frac{x_1}{\sqrt{n}}\right) \cdots g\left(\frac{x_n}{\sqrt{n}}\right),$$ where $x_1, \dots, x_n$ are freely independent self-adjoint operators with common variance $σ^2$ and $g \colon \mathbb{R} \to \mathbb{C}$ satisfies certain regularity and integrability conditions. We quantify the deviation of the singular value distribution of $π_n^{g,x}$ from the free positive semicircular law, with bounds depending only on the moments of the underlying variables. Additionally, we present a combinatorial proof of the free multiplicative CLT that extends to the unbounded setting.

math.OA↗

arXiv2024

Operator-Valued Infinitesimal Multiplicative Convolutions

We consider the notions of operator-valued infinitesimal (OVI) free independence, OVI Boolean independence, and OVI monotone independence. For each notion of OVI independence, we introduce the corresponding infinitesimal transforms, and then we show that the transforms satisfy certain multiplicative property. Additionally, we extend the concept of $t$-coefficients to the infinitesimal framework and investigate its properties. Finally, we present an application involving complex Wishart matrices utilizing our infinitesimal free multiplicative formula.

math.OA↗

arXiv2024

Infinitesimal Operators and the Distribution of Anticommutators and Commutators

In an infinitesimal probability space we consider operators which are infinitesimally free and one of which is infinitesimal, in that all its moments vanish. Many previously analysed random matrix models are captured by this framework. We show that there is a simple way of finding non-commutative distributions involving infinitesimal operators and apply this to the commutator and anticommutator. We show the joint infinitesimal distribution of an operator and an infinitesimal idempotent gives us the Boolean cumulants of the given operator. We also show that Boolean cumulants can be expressed as infinitesimal moments thus giving matrix models which exhibit asymptotic Boolean independence and monotone independence. Finally we demonstrate a connection to the Markov-Krein transform.

math.OA↗

arXiv2024

The Distribution of Polynomials in Monotone Independent Elements

Building on the work of Arizmendi and Celestino (2021), we derive the $*$-distributions of polynomials in monotone independent and infinitesimally monotone independent elements. For non-zero complex numbers $α$ and $β$, we derive explicitly the $*$-distribution of $p_{α,β}=αab + βba$ whenever $a$ and $b$ are monotone or infinitesimally monotone independent elements. This encompasses both cases of the commutator and anti-commutator. This approach can be pushed to study more general polynomials. As applications, we derive the limiting distribution with respect to the partial trace of polynomials in a certain class of random matrices.

math.PR↗

arXiv2023

A Unified approach to Infinitesimal Freeness with Amalgamation

We consider the infinitesimal freeness in the operator-valued framework, and we show that the operator-valued infinitesimal (OVI) free independence is equivalent to the operator-valued free independence over an algebra of $2\times 2$ upper triangular matrices. We introduce the notion of OVI cumulants and investigate its properties, and we then deduce that the OVI freeness is equivalent to the vanishing of our mixed cumulants. Moreover, we derive the formula for obtaining the free additive and multiplicative convolutions within the realm of OVI freeness.

math.OA↗

arXiv2022

Quantitative Estimates for Operator-Valued and Infinitesimal Boolean and Monotone Limit Theorems

We provide Berry-Esseen bounds for sums of operator-valued Boolean and monotone independent variables, in terms of the first moments of the summands. Our bounds are on the level of Cauchy transforms as well as the Lévy distance. As applications, we obtain quantitative bounds for the corresponding CLTs, provide a quantitative "fourth moment theorem" for monotone independent random variables including the operator-valued case, and generalize the results by Hao and Popa on matrices with Boolean entries. Our approach relies on a Lindeberg method that we develop for sums of Boolean/monotone independent random variables. Furthermore, we push this approach to the infinitesimal setting to obtain the first quantitative estimates for the operator-valued infinitesimal free, Boolean and monotone CLT.

math.PR↗

arXiv2021

On operator-valued infinitesimal Boolean and monotone independence

We introduce the notion of operator-valued infinitesimal (OVI) independence for the Boolean and monotone cases. Then show that OVI Boolean (resp. monotone) independence is equivalent to the operator-valued Boolean (resp. monotone) independence over an algebra of $2\times 2$ upper triangular matrices. Moreover, we derive formulas to obtain the OVI Boolean (resp. monotone) additive convolution by reducing it to the operator-valued case. We also define OVI Boolean and monotone cumulants and study its basic properties. Moreover, for each notion of OVI independence, we construct the corresponding OVI Central Limit Theorem. The relations among free, Boolean and monotone cumulants are extended to this setting. Besides, in the Boolean case we deduce that the vanishing of mixed cumulants is still equivalent to independence, and use this to connect scalar-valued with matrix-valued infinitesimal Boolean independence. Finally we study two random matrix models that are asymptotically Boolean independent but turn out to not be infinitesimally Boolean independent.

math.OA↗