arXiv subjects

Mingchu Gao

Publications and source records attributed to Mingchu Gao.

9 recordsLinked to original sources

arXiv2019

$R$-diagonal and $η$-diagonal Pairs of Random Variables

This paper is devoted to studying $R$-diagonal and $η$-diagonal pairs of random variables. We generalize circular elements to the bi-free setting, defining bi-circular element pairs of random variables, which provide examples of $R$-diagonal pairs of random variables. Formulae are given for calculating the distributions of the product pairs of two $*$-bi-free $R$-diagonal pairs. When focusing on pairs of left acting operators and right acting operators from finite von Neumann algebras in the standard form, we characterize $R$-diagonal pairs in terms of the $*$-moments of the random variables, and of distributional invariance of the random variables under multiplication by free unitaries. We define $η$-diagonal pairs of random variables, and give a characterization of $η$-diagonal pairs in terms of the $*$-distributions of the random variables. If every non-zero element in a $*$-probability space has a non-zero $*$-distribution, we prove that the unital algebra generated by a $2\times 2$ off-diagonal matrix with entries of a non-zero random variable $x$ and its adjoint $x^*$ in the algebra and the diagonal $2\times 2$ scalar matrices can never be Boolean independent fromm the $2\times 2$ scalar matrix algebra with amalgamation over the diagonal scalar matrix algebra.

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arXiv2019

Compound Bi-free Poisson Distributions

In this paper, we study compound bi-free Poisson distributions for {\sl two-faced families of random variables}. We prove a Poisson limit theorem for compound bi-free Poisson distributions. Furthermore, a bi-free infinitely divisible distribution for a two-faced family of self-adjoint random variables can be realized as the limit of a sequence of compound bi-free Poisson distributions of two-faced families of self-adjoint random variables. If a compound bi-free Poisson distribution is determined by a positive number and the distribution of a two faced family of finitely many random variables, which has an almost sure random matrix model, and the left random variables commute with the right random variables in the two-faced family, then we can construct a random bi-matrix model for the compound bi-free Poisson distribution. If a compound bi-free Poisson distribution is determined by a positive number and the distribution of a commutative pair of random variables, we can construct an asymptotic bi-matrix model with entries of creation and annihilation operators for the compound bi-free Poisson distribution.

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arXiv2019

On Bi-free Multiplicative Convolution

In this paper, we study the partial bi-free $S$-transform of a pair $(a,b)$ of random variables, and the $S$-transform of the $2\times 2$ matrix-valued random variable $\left(\begin{matrix}a&0\\0&b\end{matrix}\right)$ associated with $(a,b)$ when restricted to upper triangular $2\times 2$ matrices. We first derive an explicit expression of bi-free multiplicative convolution (of probability measures on the bi-unit-sphere $\mathbb{T}^2$ of $\mathbb{C}^2$, or on $\mathbb{R}^2_+$ in $\mathbb{C}^2$) from a subordination equation for bi-free multiplicative convolution. We then show that, when $(a_1, b_1)$ and $(a_2,b_2)$ are bi-free, the $S$-transforms of $X_1=\left(\begin{matrix}a_1&0\\0&b_1\end{matrix}\right)$, $X_2=\left(\begin{matrix}a_2&0\\0&b_2\end{matrix}\right)$ satisfy Dykema's twisted multiplicative equation for free operator-valued random variables if and only if at least one of the two partial bi-free $S$-transforms of the pairs of random variables is the constant function 1 in a neighborhood of $(0,0)$. This is the case if and only if one of the two pairs, say $(a_1,b_1)$, has factoring two-band moments (that is, $φ(a_1^mb_1^n)=φ(a_1^m)φ(b_1^n)$, for all $m,n=1, 2, \cdots$). We thus find tons of bi-free pairs of random variables to which the $S$-transforms of the corresponding matrix-value random variables do not satisfy Dykema's twisted multiplicative formula. Finally, if both $(a_1,b_1)$ and $(a_2,b_2)$ have factoring two-band moments, we prove that the $Ψ$-transforms of $X_1$, $X_2$, and $X_1X_2$ satisfy a subordination equation.

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arXiv2018

Non-Gaussian Random Matrix Models for Two-faced Families of Random Variables Having Bi-free Central Limit Distributions

In this paper, we construct random two-faced families of matrices with non-Gaussian entries to approximate a two-faced family of random variables having a bi-free central limit distribution. We prove that, under modest conditions weaker than independence, a family of random two-faced family of matrices with non-Gaussian entries is asymptotically bi-free from a two-faced family of constant diagonal matrices.

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arXiv2018

Multidimensional Free Poisson Limits on Free Stochastic Integral Algebras

In this paper, we prove four-moment theorems for multidimensional free Poisson limits on free Wigner chaos or the free Poisson algebra. We prove that, under mild technical conditions, a bi-indexed sequence of free stochastic integrals in free Wigner algebra or free Poisson algebra converges to a free sequence of free Poisson random variables if and only if the moments with order not greater than four of the sequence converge to the corresponding moments of the limit sequence of random variables. Similar four-moment theorems hold when the limit sequence is not free, but has a multidimensional free Poisson distribution with parameters $λ>0$ and $α=\{α_i: 0\ne α_i\in \mathbb{R}, i=1, 2, \cdots\}$.

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arXiv2017

Multidimensional Compound Poisson Distributions in Free Probability

Inspired by R. Speicher's multidimensional free central limit theorem and semicircle families, we prove an infinite dimensional compound Poisson limit theorem in free probability, and define infinite dimensional compound free Poisson distributions in a non-commutative probability space. Infinite dimensional free infinitely divisible distributions are defined and characterized in terms of its free cumulants. It is proved that for a distribution of a sequence of random variables, the following statements are equivalent. (1) The distribution is multidimensional free infinitely divisible. (2) The distribution is the limit distribution of triangular trays of families of random variables. (3) The distribution is the distribution of $\{a_1^{(i)}: i=1, 2, \cdots\}$ of a multidimensional free Levy process $\{\{a_t^{(i)}:i=1, 2, \cdots\}: t\ge 0\}$. (4) The distribution is the limit distribution of a sequence of multidimensional compound free Poisson distributions.

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arXiv2016

Two-faced Families of Non-commutative Random Variables Having Bi-free Infinitely Divisible Distributions

We study two-faced families of random variables having bi-free infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced pairs of random variables within a triangular array. Then, by using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free Levy processes.

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arXiv2015

Poisson Processes in Free Probability

We prove a multidimensional Poisson limit theorem in free probability, and define joint free Poisson distributions in a non-commutative probability space. We define (compound) free Poisson process explicitly, similar to the definitions of (compound) Poisson processes in classical probability. We proved that the sum of finitely many freely independent compound free Poisson processes is a compound free Poisson processes. We give a step by step procedure for constructing a (compound) free Poisson process. A Karhunen-Loeve expansion theorem for centered free Poisson processes is proved. We generalize free Poisson processes to a notion of free Poisson random measures (which is slightly different from the previously defined ones in free probability, but more like an analogue of classical Poisson random measures). Then we develop the integration theory of real-valued functions with respect to a free Poisson random measure, generalizing the classical integration theory to the free probability case. We find that the integral of a function (in certain spaces of functions) with respect to a free Poisson random measure has a compound free Poisson distribution. For centered free Poisson random measures, we can get a simpler and more beautiful integration theory.

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arXiv2010

The Relative Weak Asymptotic Homomorphism Property for Inclusions of Finite von Neumann Algebras

A triple of finite von Neumann algebras $B\subseteq N\subseteq M$ is said to have the relative weak asymptotic homomorphism property if there exists a net of unitary operators $\{u_λ\}_{λ\in Λ}$ in $B$ such that $$\lim_λ|\mathbb{E}}_B(xu_λy)-{\mathbb{E}}_B({\mathbb{E}}_N(x)u_λ{\mathbb{E}}_N(y))\|_2=0$$ for all $x,y\in M$. We prove that a triple of finite von Neumann algebras $B\subseteq N\subseteq M$ has the relative weak asymptotic homomorphism property if and only if $N$ contains the set of all $x\in M$ such that $Bx\subseteq \sum_{i=1}^n x_iB$ for a finite number of elements $x_1,...,x_n$ in $M$. Such an $x$ is called a one sided quasi-normalizer of $B$, and the von Neumann algebra generated by all one sided quasi-normalizers of $B$ is called the one sided quasi-normalizer algebra of $B$. We characterize one sided quasi-normalizer algebras for inclusions of group von Neumann algebras and use this to show that one sided quasi-normalizer algebras and quasi-normalizer algebras are not equal in general. We also give some applications to inclusions $L(H)\subseteq L(G)$ arising from containments of groups. For example, when $L(H)$ is a masa we determine the unitary normalizer algebra as the von Neumann algebra generated by the normalizers of $H$ in $G$.

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