arXiv subjects

Jiun-Chau Wang

Publications and source records attributed to Jiun-Chau Wang.

At least 19 recordsLinked to original sources

arXiv2022

Regularity results for free Lévy processes

Given a free additive convolution semigroup $\left(μ_t\right)_{t\geq 0}$ and a probability measure $ν$ on $\mathbb{R}$, we find the necessary and sufficient conditions for the process $μ_t \boxplus ν$ to be Lebesgue absolutely continuous with a positive and analytic density throughout $\mathbb{R}$ at all time $t>0$. For semigroups without this property, we find the necessary and sufficient conditions for the density of $μ_t \boxplus ν$ to be analytic at its zeros. These results are quantified by the Lévy measure of the semigroup, making it fairly easy to construct many concrete examples. Finally, we show that $μ_t \boxplus ν$ has a finite number of connected components in its support if both the Lévy measure of $\left(μ_t\right)_{t \geq 0}$ and the initial law $ν$ do.

math.PR↗

arXiv2022

Superconvergence in free probability limit theorems for arbitrary triangular arrays

It is known that limit theorems for triangular arrays with identically distributed rows yields convergence of densities rather than just convergence in distribution. We show that this superconvergence result holds -- at least at points at which the limit density is nonzero -- even if the rows of the array are not identically distributed.

math.PR↗

arXiv2021

Superconvergence and regularity of densities in free probability

The superconvergence phenomenon is shown for products of free, identically distributed random variables. We also show that a certain Holder regularity, first demonstrated by Biane for the density of a free additive convolution with a semicircular law, extends to free additive and multiplicative convolutions with arbitrary freely infinitely divisible laws and to free convolution semigroups.

math.FA↗

arXiv2020

Log-unimodality for free positive multiplicative Brownian motion

We prove that the marginal law $σ_{t}\boxtimesν$ of free positive multiplicative Brownian motion is log-unimodal for all $t>0$ if $ν$ is a multiplicatively symmetric log-unimodal distribution, and that $σ_{t}\boxtimesν$ is log-unimodal for sufficiently large $t$ if $ν$ is supported on a suitably chosen finite interval. Counterexamples are given when $ν$ is not assumed to be symmetric or having a bounded support.

math.PR↗

arXiv2018

Berry-Esseen type estimate and return sequence for parabolic iteration in the upper half-plane

Two different aspects of parabolic iteration in the complex upper half-plane are considered here. First, from a noncommutative probability perspective, a Berry-Esseen type estimate for the convergence speed of the monotone central limit theorem is proved. Secondly, if the underlying measure in this central limit process is singular to the Lebesgue measure on the real line, then the iteration is shown to be an infinite-measure preserving dynamical system that has a regularly varying return sequence of index 1/2.

math.FA↗

arXiv2018

Bi-Free Extreme Values

In this paper, we continue Voiculescu's recent work on the analogous extreme value theory in the context of bi-free probability theory. We derive various equivalent conditions for a bivariate distribution function to be bi-freely max-infinitely divisible. A bi-freely max-infinitely divisible distribution function can be expressed in terms of its marginals and a special form of copulas. Such a distribution function is shown to be also max-infinitely divisible in the classical sense. In addition, we characterize the set of bi-free extreme value distribution functions. A distribution function of this type is also bi-free max-stable and represented by its marginals and one copula composing of a Pickands dependence function, as in the classical extreme value theory. As a consequence, the determination of its bi-free domain of attraction is the same as the criteria in the classical theory. To illustrate these connections, some concrete examples are provided.

math.OA↗

arXiv2017

Limit theorems in bi-free probability theory

In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of selfadjoint commuting random variables in an infinitesimal triangular array are determined. These distributions are characterized by their bi-freely infinite divisibility, and moreover, a transfer principle is established for limit theorems in classical probability theory and Voiculescu's bi-free probability theory. Complete descriptions of bi-free stability and fullness of planar probability distributions are also set down. All these results reveal one important feature about the theory of bi-free probability that it parallels the classical theory perfectly well. The emphasis in the whole work is not on the tool of bi-free combinatorics but only on the analytic machinery.

math.PR↗

arXiv2017

Harmonic analysis for the bi-free partial S-transform

This is a continuation of our previous work in bi-free harmonic analysis for commuting left and right variables. Here we analyze the bi-free partial S-transform and use the results to study limit theorems and infinite divisibility relative to the multiplicative bi-free convolution.

math.OA↗

arXiv2016

On the multiplication of operator-valued c-free random variables

We discuss some results concerning the multiplication of non-commutative random variables that are c-free with respect to a pair $( Φ, φ) $, where $ Φ$ is a linear map with values in some Banach or C$^\ast$-algebra and $ φ$ is scalar-valued. In particular, we construct a suitable analogue of the Voiculescu's $ S $-transform for this framework.

math.OA↗

arXiv2016

Analytic aspects of the bi-free partial R-transform

Since Voiculescu introduced his bi-free probability theory in 2013, the major development of the theory has been on its combinatorial side; in particular, on the combinatorics of bi-free cumulants and its application to the bi-free R-transform. In this article we propose a harmonic analysis approach to the bi-free R-transform, which is solely based on integral transforms of two variables. To accommodate the harmonic analysis tools, we confine ourselves in the simplest situation of bi-freeness with commuting faces. Our method allows us to treat measures with unbounded support, and we show that the classical limit theory of infinitely divisible laws, due to Levy and Khintchine, has a perfect bi-free analogue.

math.OA↗

arXiv2015

Superconvergence to freely infinitely divisible distributions

The phenomenon of superconvergence is proved for all freely infinitely divisible distributions. Precisely, suppose that the partial sums of a sequence of free identically distributed, infinitesimal random variables converge in distribution to a nondegenerate freely infinitely divisible law. Then the distribution of the sum becomes Lebesgue absolutely continuous with a continuous density in finite time, and this density can be approximated by that of the limit law uniformly, as well as in all $L^{p}$-norms for $p>1$, on the real line except possibly in the neighborhood of one point. Applications include the global superconvergence to freely stable laws and that to free compound Poisson laws over the whole real line.

math.PR↗

arXiv2014

Local limit theorems for multiplicative free convolutions

This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the superconvergence to free Brownian motion and that to the Haar measure are shown to be uniform over the entire unit circle, implying further a free entropic limit theorem and a universality result for unitary free Lévy processes. Finally, the method of proofs on the positive half-line gives rise to a new multiplicative Boolean to free Bercovici-Pata bijection.

math.FA↗

arXiv2013

The central limit theorem for monotone convolution with applications to free Levy processes and infinite ergodic theory

In this paper free harmonic analysis tools are used to study parabolic iteration in the complex upper half-plane. The main result here is a complete characterization for the norming constants in the monotonic central limit theorem. This allows us to construct a new class of conservative and ergodic measure-preserving transformations on the real line with Lebesgue measure. Among all, we mention that the generalized Boole transformation with infinitely many poles is shown to be conservative, as long as its residues are summable.

math.FA↗

arXiv2010

Local limit theorems in free probability theory

In this paper, we study the superconvergence phenomenon in the free central limit theorem for identically distributed, unbounded summands. We prove not only the uniform convergence of the densities to the semicircular density but also their $L^p$-convergence to the same limit for $p>1/2$. Moreover, an entropic central limit theorem is obtained as a consequence of the above results.

math.PR↗

arXiv2008

On multiplicative conditionally free convolution

Using the combinatorics of non-crossing partitions, we construct a conditionally free analogue of the Voiculescu's S-transform. The result is applied to analytical description of conditionally free multiplicative convolution and characterization of infinite divisibility.

math.OA↗

arXiv2008

Limit theorems for additive c-free convolution

In this paper we find necessary and sufficient conditions for the weak convergence of c-free convolution of pairs of measures, where the measures are assumed to be infinitesimal and their support may be unbounded. These results are obtained by complex analytic methods.

math.OA↗

arXiv2007

On freely indecomposable measures

We show that a probability measure is not a nontrivial free additive convolution if it puts no mass in an interval whose endpoints are atoms. The analogous results for free multiplicative convolutions are proved as well. The proofs use analytic subordination.

math.OA↗

arXiv2007

Limit laws for boolean convolutions

We study the distributional behavior for products, and for sums of boolean independent random variables in an infinitesimal triangular array. We show that the limit laws of boolean convolutions are determined by the limit laws of free convolutions, and vice versa. We further use these results to show several connections between the limiting distributional behavior of classical convolutions and that of boolean convolutions. The proof of our results is based on the analytical apparatus developed for free convolutions.

math.OA↗