arXiv subjects

Makoto Maejima

Publications and source records attributed to Makoto Maejima.

16 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↗

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↗

arXiv2014

On the range of exponential functionals of Lévy processes

We characterize the support of the law of the exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$ of two one-dimensional independent Lévy processes $ξ$ and $η$. Further, we study the range of the mapping $Φ_ξ$ for a fixed Lévy process $ξ$, which maps the law of $η_1$ to the law of the corresponding exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$. It is shown that the range of this mapping is closed under weak convergence and in the special case of positive distributions several characterizations of laws in the range are given.

math.PR↗

arXiv2013

Stochastic integral and series representations for strictly stable distributions

In this paper we find and develop a stochastic integral representation for the class of strictly stable distributions. We establish an explicit relationship between stochastic integral and shot-noise series representations of strictly stable distributions, which shows that the class of distributions representable by stochastic integral is larger than the class representable by a shot-noise series. This inclusion is proper when the stability index is greater than 1. We also give an explicit description of distributions possessing both representations.

math.PR↗

arXiv2012

Distributions of exponential integrals of independent increment processes related to generalized gamma convolutions

It is known that in many cases distributions of exponential integrals of Levy processes are infinitely divisible and in some cases they are also selfdecomposable. In this paper, we give some sufficient conditions under which distributions of exponential integrals are not only selfdecomposable but furthermore are generalized gamma convolution. We also study exponential integrals of more general independent increment processes. Several examples are given for illustration.

math.ST↗

arXiv2012

The dichotomy of recurrence and transience of semi-Levy processes

Semi-Levy process is an additive process with periodically stationary increments. In particular, it is a generalization of Levy process. The dichotomy of recurrence and transience of Levy processes is well known, but this is not necessarily true for general additive processes. In this paper, we prove the recurrence and transience dichotomy of semi-Levy processes. For the proof, we introduce a concept of semi-random walk and discuss its recurrence and transience properties. An example of semi-Levy process constructed from two independent Levy processes is investigated. Finally, we prove the laws of large numbers for semi-Levy processes.

math.PR↗

arXiv2012

A class of multivariate infinitely divisible distributions related to arcsine density

Two transformations $\mathcal{A}_1$ and $\mathcal{A}_2$ of Lévy measures on $\mathbb{R}^d$ based on the arcsine density are studied and their relation to general Upsilon transformations is considered. The domains of definition of $\mathcal{A}_1$ and $\mathcal{A}_2$ are determined and it is shown that they have the same range. The class of infinitely divisible distributions on $\mathbb{R}^d$ with Lévy measures being in the common range is called the class $A$ and any distribution in the class $A$ is expressed as the law of a stochastic integral $\int_0^1\cos(2^{-1}\uppi t)\,\mathrm{d}X_t$ with respect to a Lévy process $\{X_t\}$. This new class includes as a proper subclass the Jurek class of distributions. It is shown that generalized type $G$ distributions are the image of distributions in the class $A$ under a mapping defined by an appropriate stochastic integral. $\mathcal{A}_2$ is identified as an Upsilon transformation, while $\mathcal{A}_1$ is shown not to be.

math.ST↗

arXiv2010

Type A Distributions: Infinitely Divisible Distributions Related to Arcsine Density

Two transformations $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ of Lévy measures on $\mathbb{R}^{d}$ based on the arcsine density are studied and their relation to general Upsilon transformations is considered. The domains of definition of $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are determined and it is shown that they have the same range. Infinitely divisible distributions on $\mathbb{R}^{d}$ with Lévy measures being in the common range are called type $A$ distributions and expressed as the law of a stochastic integral $\int_0^1\cos (2^{-1}πt)dX_t$ with respect to Lévy process $\{X_t\}$. \ This new class includes as a proper subclass the Jurek class of distributions. It is shown that generalized type $G$ distributions are the image of type $A$ distributions under a mapping defined by an appropriate stochastic integral. $\mathcal{A}_{2}$ is identified as an Upsilon transformation, while $\mathcal{A}_{1}$ is shown to be not.

math.PR↗

arXiv2010

Nested subclasses of the class of $α$-selfdecomposable distributions

A probability distribution $μ$ on $\mathbb R ^d$ is selfdecomposable if its characteristic function $\widehatμ(z), z\in\mathbb R ^d$, satisfies that for any $b>1$, there exists an infinitely divisible distribution $ρ_b$ satisfying $\widehatμ(z) = \widehatμ(b^{-1}z)\widehatρ_b(z)$. This concept has been generalized to the concept of $α$-selfdecomposability by many authors in the following way. Let $α\in\mathbb R$. An infinitely divisible distribution $μ$ on $\mathbb R ^d$ is $α$-selfdecomposable, if for any $b>1$, there exists an infinitely divisible distribution $ρ_b$ satisfying $\widehatμ(z) = \widehat μ(b^{-1}z)^{b^α}\widehatρ_b(z)$. By denoting the class of all $α$-selfdecomposable distributions on $\mathbb R ^d$ by $L^{\leftangleα\rightangle}(\mathbb R ^d)$, we define in this paper a sequence of nested subclasses of $L^{\leftangleα\rightangle}(\mathbb R ^d)$, and investigate several properties of them by two ways. One is by using limit theorems and the other is by using mappings of infinitely divisible distributions.

math.PR↗

arXiv2009

Stochastic integral characterizations of semi-selfdecomposable distributions and related Ornstein-Uhlenbeck type processes

In this paper, three topics on semi-selfdecomposable distributions are studied. The first one is to characterize semi-selfdecomposable distributions by stochastic integrals with respect to Levy processes. This characterization defines a mapping from an infinitely divisible distribution with finite log-moment to a semi-selfdecomposable distribution. The second one is to introduce and study a Langevin type equation and the corresponding Ornstein-Uhlenbecktype process whose limiting distribution is semi-selfdecomposable. Also, semi-stationary Ornstein-Uhlenbeck type processes with semi-selfdecomposable distributions are constructed. The third one is to study the iteration of the mapping above. The iterated mapping is expressed as a single mapping with a different integrand. Also, nested subclasses of the class of semi-selfdecomposable distributions are considered, andit is shown that the limit of these nested subclasses is the closure of the class of semi-stable distributions.

math.PR↗

arXiv2009

New Classes of Infinitely Divisible Distributions Related to the Goldie-Steutel-Bondesson Class

Recently, many classes of infinitely divisible distributions on R^d have been characterized in several ways. Among others, the first way is to use Levy measures, the second one is to use transformations of Levy measures, and the third one is to use mappings of infinitely divisible distributions defined by stochastic integrals with respect to Levy processes. In this paper, we are concerned with a class of mappings, by which we construct new classes of infinitely divisible distributions on R^d. Then we study a special case in R^1, which is the class of infinitely divisible distributions without Gaussian parts generated by stochastic integrals with respect to a fixed compound Poisson processes on R^1. This is closely related to the Goldie-Steutel-Bondesson class.

math.PR↗

arXiv2009

Classes of infinitely divisible distributions on R^d related to the class of selfdecomposable distributions

This paper studies new classes of infinitely divisible distributions on R^d. Firstly, the connecting classes with a continuous parameter between the Jurek class and the class of selfdecomposable distributions are revisited. Secondly, the range of the parameter is extended to construct new classes and characterizations in terms of stochastic integrals with respect to Levy processes are given. Finally, the nested subclasses of those classes are discussed and characterized in two ways: One is by stochastic integral representations and another is in terms of Levy measures.

math.PR↗

arXiv2008

Limits of bifractional Brownian noises

Let $B^{H,K}=(B^{H,K}_{t}, t\geq 0)$ be a bifractional Brownian motion with two parameters $H\in (0,1)$ and $K\in(0,1]$. The main result of this paper is that the increment process generated by the bifractional Brownian motion $(B^{H,K}_{h+t} -B^{H,K}_{h}, t\geq 0)$ converges when $h\to \infty$ to $(2^{(1-K)/{2}}B^{HK}_{t}, t\geq 0)$, where $(B^{HK}_{t}, t\geq 0)$ is the fractional Brownian motion with Hurst index $HK$. We also study the behavior of the noise associated to the bifractional Brownian motion and limit theorems to $B^{H,K}$.

math.PR↗

arXiv2007

The limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions

It is shown that the limits of the nested subclasses of five classes of infinitely divisible distributions on $R^d$, which are the Jurek class, the Goldie-Steutel-Bondesson class, the class of selfdecomposable distributions, the Thorin class and the class of generalized type $G$ distributions, are identical with the closure of the class of stable distributions. More general results are also given.

math.PR↗

arXiv2006

Some properties of exponential integrals of Lévy processes and examples

The improper stochastic integral $Z=\int_0^{\infty-}\exp(-X_{s-})dY_s$ is studied, where $\{(X_t, Y_t), t \geqslant 0 \}$ is a Lévy process on $\mathbb R ^{1+d}$ with $\{X_t \}$ and $\{Y_t \}$ being $\mathbb R$-valued and $\mathbb R ^d$-valued, respectively. The condition for existence and finiteness of $Z$ is given and then the law $\mathcal L(Z)$ of $Z$ is considered. Some sufficient conditions for $\mathcal L(Z)$ to be selfdecomposable and some sufficient conditions for $\mathcal L(Z)$ to be non-selfdecomposable but semi-selfdecomposable are given. Attention is paid to the case where $d=1$, $\{X_t\}$ is a Poisson process, and $\{X_t\}$ and $\{Y_t\}$ are independent. An example of $Z$ of type $G$ with selfdecomposable mixing distribution is given.

math.PR↗