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Noriyoshi Sakuma

Publications and source records attributed to Noriyoshi Sakuma.

At least 19 recordsLinked to original sources
arXiv2026

The Berry--Esseen Estimate in the Free Central Limit Theorem

We consider sums of freely independent self-adjoint random variables that are not necessarily identically distributed. Let $μ_j$ denote the distribution of the $j$th summand. We assume that they have mean zero and finite absolute moments of order $2+δ$, where $0<δ\le 1$. Let $Δ$ denote the Kolmogorov distance, let $μ^{(n)}$ be the distribution of the normalized partial sum, let $ω$ be the standard semicircle law, and let $B_n^2$ be the variance of the partial sum. The purpose of this paper is to prove the Berry--Esseen estimate in the free central limit theorem. Namely, there exists an absolute constant $C>0$ such that, for every $0<δ\le 1$, \[ Δ(μ^{(n)},ω) \le \frac{C}{B_n^{2+δ}}\sum_{j=1}^n \int_{\R}|x|^{2+δ}\,μ_j(dx), \] Our result not only improves several known estimates for general non-identically distributed random variables, but also establishes exactly the same Berry--Esseen estimate as in classical probability theory. The proof combines truncation, a quantitative estimate for the $R$-transform, a stability analysis of a perturbed semicircle equation, and a Bai-type smoothing inequality.

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↗
arXiv2026

Modified ruin probability for a Cramér-Lundberg model driven by a compound mixed Poisson process

We study modified ruin probabilities in a Cramér-Lundberg model driven by a compound mixed Poisson process. In the heavy-tailed regime, if the integrated claim-size distribution is subexponential and the upper endpoint of the mixing distribution stays below the net-profit boundary, the modified and classical ruin probabilities are asymptotically equivalent. In the light-tailed regime, we prove a fixed-intensity ratio theorem and obtain both an endpoint-atom result and a sharp endpoint-density asymptotic with an explicit constant.

math.PR↗
arXiv2025

Generalized Meixner-type free gamma distributions:convolution formulas and potential correspondence

We introduce and study a class of generalized Meixner-type free gamma distributions $μ_{t,θ,λ}$ ($t,θ>0$ and $λ\ge 1$), which includes both the free gamma distributions introduced by Anshelevich and certain scaled free beta prime distributions introduced by Yoshida. We investigate fundamental properties and mixture structures of these distributions. In particular, we consider the Gibbs distribution $\frac{1}{\mathcal{Z}_{t,θ,λ}} \exp\{-V_{t,θ,λ}(x)\}$ associated with a family of potentials $V_{t,θ,λ}$, and show that $μ_{t,θ,λ}$ maximizes Voiculescu's free entropy with potential $V_{t,θ,λ}$ for parameters $t,θ>0$ and $1\le λ<1+t/θ$. This result substantially extends the range of classcal-free correspondences obtained the potential function, differing from those arising from the Bercovici-Pata bijection. Moreover, we identify algebraic relations involving noncommutative random variables distributed as free gamma distributions.

math.PR↗
arXiv2024

Fluctuations of eigenvalues of a polynomial on Haar unitary and finite rank matrices

This paper calculates the fluctuations of eigenvalues of polynomials on large Haar unitaries cut by finite rank deterministic matrices. When the eigenvalues are all simple, we can give a complete algorithm for computing the fluctuations. When multiple eigenvalues are involved, we present several examples suggesting that a general algorithm would be much more complex.

math.PR↗
arXiv2023

Rates of convergence in the free central limit theorem

We study the free central limit theorem for not necessarily identically distributed free random variables where the limiting distribution is the semicircle distribution. Starting from an estimate for the Kolmogorov distance between the measure of suitably normalized sums of free random variables and the semicircle distribution without any moment condition, we show the free Lindeberg central limit theorem and improve the known results on rates of convergence under the conditions of the existence of the third moments.

math.PR↗
arXiv2022

Selfsimilar free additive processes and freely selfdecomposable distributions

In the paper by Fan\cite{F06}, he introduced the marginal selfsimilarity of non-commutative stochastic processes and proved the marginal distributions of selfsimilar processes with freely independent increments are freely selfdecomposable. In this paper, we firstly introduce a new definition, stronger than Fan's one in general, of selfsimilarity via linear combinations of non-commutative stochastic processes, although their two definitions are equivalent for non-commutative stochastic processes with freely independent increments. We secondly prove the converse of Fan's result, to complete the relationship between selfsimilar free additive processes and freely selfdecomposable distributions. Furthermore, we construct stochastic integrals with respect to free additive processes for representing the background driving free L{é}vy processes of freely selfdecomposable distributions. A relationship between freely selfdecomposable distributions and their background driving free L{é}vy processes in terms of their free cumulant transforms is also given, and several examples are discussed.

math.PR↗
arXiv2022

Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases

We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the measure-valued processes so that their Cauchy transforms solve the complex Burgers-type equations. We show that applications of the method of characteristic curves to these partial differential equations provide the functional equations relating the Cauchy transforms of measures at an arbitrary time with those at the initial time. We transform the functional equations for the Cauchy transforms to those for the $R$-transforms and the $S$-transforms of the measures, which play central roles in free probability theory. The obtained functional equations for the $R$-transforms and the $S$-transforms are simpler than those for the Cauchy transforms and useful for explicit calculations including the computation of free cumulant sequences. Some of the results are argued using the notion of free convolutions.

math.PR↗
arXiv2022

On Boolean selfdecomposable distributions

This paper introduces the class of selfdecomposable distributions concerning Boolean convolution. A general regularity property of Boolean selfdecomposable distributions is established; in particular the number of atoms is at most two and the singular continuous part is zero. We then analyze how shifting probability measures changes Boolean selfdecomposability. Several examples are presented to supplement the above results. Finally, we prove that the standard normal distribution $N(0,1)$ is Boolean selfdecomposable but the shifted one $N(m,1)$ is not for sufficiently large $|m|$.

math.PR↗
arXiv2022

On freely quasi-infinitely divisible distributions

Inspired by the notion of quasi-infinite divisibility (QID), we introduce and study the class of freely quasi-infinitely divisible (FQID) distributions on $\mathbb{R}$, i.e. distributions which admit the free Lévy-Khintchine-type representation with signed Lévy measure. We prove several properties of the FQID class, some of them in contrast to those of the QID class. For example, a FQID distribution may have negative Gaussian part, and the total mass of its signed Lévy measure may be negative. Finally, we extend the Bercovici-Pata bijection, providing a characteristic triplet, with the Lévy measure having nonzero negative part, which is at the same time classical and free characteristic triplet.

math.PR↗
arXiv2022

Matrix models for cyclic monotone and monotone independences

Cyclic monotone independence is an algebraic notion of noncommutative independence, introduced in the study of multi-matrix random matrix models with small rank. Its algebraic form turns out to be surprisingly close to monotone independence, which is why it was named cyclic monotone independence. This paper conceptualizes this notion by showing that the same random matrix model is also a model for the monotone convergence with an appropriately chosen state. This observation provides a unified nonrandom matrix model for both types of monotone independences.

math.OA↗
arXiv2017

The normal distribution is freely selfdecomposable

The class of selfdecomposable distributions in free probability theory was introduced by Barndorff-Nielsen and the third named author. It constitutes a fairly large subclass of the freely infinitely divisible distributions, but so far specific examples have been limited to Wigner's semicircle distributions, the free stable distributions, two kinds of free gamma distributions and a few other examples. In this paper, we prove that the (classical) normal distributions are freely selfdecomposable. More generally it is established that the Askey-Wimp-Kerov distribution $μ_c$ is freely selfdecomposable for any $c$ in $[-1,0]$. The main ingredient in the proof is a general characterization of the freely selfdecomposable distributions in terms of the derivative of their free cumulant transform.

math.PR↗
arXiv2017

Free probability for purely discrete eigenvalues of random matrices

In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescu's sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model.

math.PR↗
arXiv2016

Unimodality for free Lévy processes

We will prove that: (1) A symmetric free Lévy process is unimodal if and only if its free Lévy measure is unimodal; (2) Every free Lévy process with boundedly supported Lévy measure is unimodal in sufficiently large time. (2) is completely different property from classical Lévy processes. On the other hand, we find a free Lévy process such that its marginal distribution is not unimodal for any time $s>0$ and its free Lévy measure does not have a bounded support. Therefore, we conclude that the boundedness of the support of free Lévy measure in (2) cannot be dropped. For the proof we will (almost) characterize the existence of atoms and the existence of continuous probability densities of marginal distributions of a free Lévy process in terms of Lévy--Khintchine representation.

math.PR↗
arXiv2013

New limit theorems related to free multiplicative convolution

Let $\boxplus$, $\boxtimes$ and $\uplus$ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure $μ$ on $[0,\infty)$ with finite second moment, we find the scaling limit of $(μ^{\boxtimes N})^{\boxplus N}$ as $N$ goes to infinity. The $\mathcal{R}$--transform of the limit distribution can be represented by the Lambert's $W$ function. We also find similar limit theorem by replacing the free additive convolution with the boolean convolution.

math.PR↗
arXiv2012

On the Law of Free Subordinators

We study the freely infinitely divisible distributions that appear as the laws of free subordinators. This is the free analog of classically infinitely divisible distributions supported on [0,\infty), called the free regular measures. We prove that the class of free regular measures is closed under the free multiplicative convolution, t-th boolean power for $0\leq t\leq 1$, t-th free multiplicative power for $t\geq 1$ and weak convergence. In addition, we show that a symmetric distribution is freely infinitely divisible if and only if its square can be represented as the free multiplicative convolution of a free Poisson and a free regular measure. This gives two new explicit examples of distributions which are infinitely divisible with respect to both classical and free convolutions: χ^2(1) and F(1,1). Another consequence is that the free commutator operation preserves free infinite divisibility.

math.PR↗