Evgeny Burnaev
The aim of this note is to prove the inversion formula, which can be used to compute the Levi measure of an infinitely divisible distribution from its characteristic function. Obtained formula is similar to the well-known inversion formula [2], which is used to compute the distribution function of a random variable from the corresponding characteristic function.
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Infinitely divisible cylindrical measures on Banach spaces
本文引入了任意巴拿赫空间上的无限可分圆柱概率测度。一类无限可分的圆柱概率测度的描述，在他们的特点，这是不知道在一般情况下无限可分的Radon措施巴拿赫空间的特征。推导出无限可分柱测度的进一步性质，例如连续性。此外，分类的结果使我们能够得到关于巴拿赫空间上真Levy测度的新结果.
Ergodic Properties of Max-Infinitely Divisible Processes
我们证明了一个平稳的最大-无限可分过程是混合（遍历）当且仅当它的依赖函数收敛到0（是Cesaro可和到0）。这些准则适用于某些类极大无限可分过程。
Infinite divisibility of random fields admitting an integral representation with an infinitely divisible integrator
考虑可表示为确定性函数关于无穷可分随机测度的积分的随机场，并证明这些随机场是无穷可分的。
Some results on random fields admitting a spectral representation with infinitely divisible integrator
本文研究了具有无穷可分积分的谱表示的随机场，并证明了它们的一些性质。
Let $Z$ be a standard normal random variable (r.v.). It is shown that the distribution of the r.v. $\ln|Z|$ is infinitely divisible; equivalently, the standard normal distribution considered as the distribution on the multiplicative group over $\mathbb{R}\setminus\{0\}$ is infinitely divisible.
On Critical Points for Gaussian Vectors with Infinitely Divisible Squares
研究了零均值三维高斯向量的无穷可分平方的扰动大小，使其无穷可分性保持不变。
Based on the~method of subordinating functions we prove bounds for the minimal error of approximations of $n$-fold convolutions of probability measures by free infinitely divisible probability measures.
Random matrix models of stochastic integral type for free infinitely divisible distributions
Bercovici-Pata双射将经典的无穷可分分布的集合映射到自由的无穷可分分布的集合。本文研究了在这种双射下自由无穷可分分布的随机矩阵模型。首先，我们找到了一个特殊形式的极式分解的Lévy措施的随机矩阵模型考虑Benaych-Georges谁介绍了模型，通过他们的措施。其次，当自由无穷可分分布的经典无穷可分分布具有随机整表示时，建立了由无限可分矩阵随机积分组成的随机矩阵模型.这些随机矩阵模型是由关于矩阵值Lévy过程的随机积分给出的随机矩阵的实现。给出了几类自由无穷可分分布的随机矩阵模型的例子。特别地，证明了任意自由自分解无穷可分分布都有Ornstein-Uhlenbeck型随机矩阵模型$\int_{0}^{\infty}e^{-t}dΨ_{t}^{d}$，$d\geq1$，其中$Ψ_{t}^{d}$是满足$I_{\log}$条件的$d\times d$矩阵值Lévy过程.
For the class of free-infinitely divisible transforms are introduced three families of increasing Urbanik type subclasses of those transforms. They begin with the class of free-normal transforms and end up with the whole class of free-infinitely divisible transforms. Those subclasses are derived from the ones of classical infinitely divisible measures for which are known their random integral representations. Special functions like Hurwitz-Lerch, polygamma and hypergeometric appeared in kernels of the corresponding integral representations.
We propose isomorphism type identities for nonlinear functionals of general infinitely divisible processes. Such identities can be viewed as an analogy of the Cameron-Martin formula for Poissonian infinitely divisible processes but with random translations. The applicability of these tools relies on a precise understanding of Lévy measures of infinitely divisible processes and their representations, which are developed here in full generality. We illustrate this approach on examples of squared Bessel processes, Feller diffusions, permanental processes, as well as Lévy processes.
We derive some estimates for the integral modulus of continuity of probability densities of infinitely divisible distributions. The paper is splitted into two parts. The first part deals with general infinitely divisible distributions. The second part is mainly concerned with densities of random integrals with respect to a Lévy process. We will see major differences between compact and non-compact supports.
A probability distribution $μ$ on $\mathbb{R}^d$ is quasi-infinitely divisible if its characteristic function has the representation $\widehatμ = \widehat{μ_1}/\widehat{μ_2}$ with infinitely divisible distributions $μ_1$ and $μ_2$. In \cite[Thm. 4.1]{lindner2018} it was shown that the class of quasi-infinitely divisible distributions on $\mathbb{R}$ is dense in the class of distributions on $\mathbb{R}$ with respect to weak convergence. In this paper, we show that the class of quasi-infinitely divisible distributions on $\mathbb{R}^d$ is not dense in the class of distributions on $\mathbb{R}^d…
Simulation of infinitely divisible random fields
给出了两种逼近无限可分随机场的方法。该方法是基于近似的核函数的谱表示的字段，导致数值积分的各自的积分。误差界的近似误差推导和近似用于模拟某些类的无限可分随机场。
A quasi-infinitely divisible distribution on $\mathbb{R}$ is a probability distribution whose characteristic function allows a Lévy-Khintchine type representation with a "signed Lévy measure", rather than a Lévy measure. Quasi-infinitely divisible distributions appear naturally in the factorization of infinitely divisible distributions. Namely, a distribution $μ$ is quasi-infinitely divisible if and only if there are two infinitely divisible distributions $μ_1$ and $μ_2$ such that $μ_1 \ast μ= μ_2$. The present paper studies certain properties of quasi-infinitely divisible distributions in ter…
The classical infinite divisibility of distributions related to eigenvalues of some random matrix ensembles is investigated. It is proved that the $β$-Tracy-Widom distribution, which is the limiting distribution of the largest eigenvalue of a $β$-Hermite ensemble, is not infinitely divisible. Furthermore, for each fixed $N \ge 2$ it is proved that the largest eigenvalue of a GOE/GUE random matrix is not infinitely divisible.
We give a necessary and sufficient condition for symmetric infinitely divisible distribution to have Gaussian component. The result can be applied to approximation the distribution of finite sums of random variables. Particularly, it shows that for a large class of distributions with finite variance stable approximation appears to be better than Gaussian. keywords: infinitely divisible distributions; Gaussian component; approximations of sums of random variables.
We show that, under the long-tailedness of the densities of normalized Lévy measures, the densities of infinitely divisible distributions on the half line are subexponential if and only if the densities of their normalized Lévy measures are subexponential. Moreover, we prove that, under a certain continuity assumption, the densities of infinitely divisible distributions on the half line are subexponential if and only if their normalized Lévy measures are locally subexponential.
We consider infinitely divisible distributions with symmetric Lévy measure and study the absolute continuity of them with respect to the Lebesgue measure. We prove that if $η(r)=\int_{|x|\le r} x^2 ν(dx)$ where $ν$ is the Lévy measure, then $\int_0^1 \frac{r}{η(r)}dr <\infty$ is a sufficient condition for absolute continuity. As far as we know, our result is not implied by existing results about absolute continuity of infinitely divisible distributions.
New Classes of Infinitely Divisible Distributions Related to the Goldie-Steutel-Bondesson Class
最近，R^d上的许多类无穷可分分布已经用几种方法刻画。其中，第一种方法是利用Levy测度，第二种方法是利用Levy测度的变换，第三种方法是利用关于Levy过程的随机积分所定义的无穷可分分布的映射.本文研究了一类映射，利用这类映射构造了R^d上新的无穷可分分布类.然后研究了R^1中的一种特殊情况，即R^1上关于固定复合Poisson过程的随机积分所生成的无高斯部分的无穷可分分布。这与Goldie-Steutel-Bondesson类密切相关。
On processes which are infinitely divisible with respect to time
这篇短文的目的是提出IDT过程的概念，IDT过程是从修改的无限可分性质得到的Lévy过程的广泛推广。特别注意的是把一些例子，以澄清有多少IDT过程要么不同，或类似于，Lévy过程。
The law of a positive infinitely divisible process with no drift is characterized by its Lévy measure on the paths space. Based on recent results of the two authors, it is shown that even for simple examples of such processes, the knowledge of their Lévy measures allows to obtain remarkable distributional identities.
Structure of infinitely divisible semimartingales
本文给出了无限可分半鞅的一个完整刻画，即，半鞅的有限维分布是无限可分的。得到了这类半鞅的一个显式且本质唯一的分解。本文提出了一种新的方法，将无穷可分过程的级数分解与其跳跃的详细分析相结合。作为主要结果的一个例证，我们明确地确定了一大类平稳增量过程的半鞅性质，并考虑了几个感兴趣的过程的例子。这些结果推广了Stricker定理描述高斯半鞅和Knight定理描述高斯移动平均半鞅，特别是。
On the embeddability of certain infinitely divisible probability measures on Lie groups
给出了一类连通李群上的无穷可分概率测度可嵌入连续单参数概率测度卷积半群的充分条件.（定理1.3）。这使我们特别能够得出所有无限可分概率测度在某些李群上的可嵌入性，包括所谓的核桃群（推论1.5）。在其他条件下也得到了可嵌入性的结论（推论1.4和定理1.6）。
Stricker's theorem states that a Gaussian process is a semimartingale in its natural filtration if and only if it is the sum of an independent increment Gaussian process and a Gaussian process of finite variation, see [1983, Z. Wahrsch. Verw. Gebiete 64(3)]. We consider extensions of this result to non Gaussian infinitely divisible processes. First we show that the class of infinitely divisible semimartingales is so large that the natural analog of Stricker's theorem fails to hold. Then, as the main result, we prove that an infinitely divisible semimartingale relative to the filtration generat…
The suprema of infinitely divisible processes
该研究针对无穷可分过程期望上确界的刻画问题，移除了$H(C_{0},δ)$技术性假设，完整解决了M. Talagrand提出的相关猜想，完善了该领域的理论结果。
Sharp asymptotics for the free energy of 1+1 dimensional directed polymers in an infinitely divisible environment
给出了1+1维无规环境中定向高聚物自由能的精确估计。这个估计是已知的高斯环境，我们将其扩展到的情况下，环境的法律是无限可分的。
A note on the gaps in the support of discretely infinitely divisible laws
本文从复合Poisson律/过程的角度讨论了离散无穷可分分布支撑下间隙的性质。本文的讨论推广到非负实直线上的无穷可分分布。
Variation comparison between infinitely divisible distributions and the normal distribution
本文证明具有有限二阶矩的随机变量满足特定概率不等式，该不等式对多种无穷可分连续分布成立，且数值结果支持离散情形。
A quasi-infinitely divisible distribution on $\mathbb{R}^d$ is a probability distribution $μ$ on $\mathbb{R}^d$ whose characteristic function can be written as the quotient of the characteristic functions of two infinitely divisible distributions on $\mathbb{R}^d$. Equivalently, it can be characterised as a probability distribution whose characteristic function has a Lévy--Khintchine type representation with a "signed Lévy measure", a so called quasi--Lévy measure, rather than a Lévy measure. A systematic study of such distributions in the univariate case has been carried out in Lindner, Pan a…
Random integral mappings $I^{h,r}_{(a,b]}$ give isomorphisms between the sub-semigroups of the classical $(ID, \ast)$ and the free-infinite divisible $(ID,\boxplus)$ probability measures. This allows us to introduce new examples of such measures and their corresponding characteristic functionals.
A characterization of the infinitely divisible squared Gaussian processes
我们证明了，直到乘以常数，高斯过程有一个无限可分的平方当且仅当其协方差是一个瞬态马尔可夫过程的格林函数。
We study two ways (levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms. For free-selfdecomposable measures we found the formula (a differential equation) for their background driving transforms. We illustrate our methods on the hyperbolic characteristic functions. As a by-product our approach may produce new formulas for some definite integrals.
Characterization of the finite variation property for a class of stationary increment infinitely divisible processes
我们刻画了由无穷可分随机测度驱动的平稳增量混合移动平均线的有限变差性质。这些过程包括由Levy过程驱动的分数过程和移动平均过程，以及它们的混合过程。对于有限变分性质，我们建立了两类零-一定律。我们还考虑了一些例子来说明我们的结果。
In this article we illustrate the relation between the existence of Wiener integrals with respect to a Levy process in a separable Banach space and radonifying operators. For this purpose, we introduce the class of theta-radonifying operators, i.e. operators which map a cylindrical measure theta to a genuine Radon measure. We study this class of operators for various examples of infinitely divisible cylindrical measures theta and highlight the differences from the Gaussian case.
The concept of a Lévy subordinator is generalized to a family of non-decreasing stochastic processes, which are parameterized in terms of two Bernstein functions. Whereas the independent increments property is only maintained in the Lévy subordinator special case, the considered family is always strongly infinitely divisible with respect to time, meaning that a path can be represented in distribution as a finite sum with arbitrarily many summands of independent and identically distributed paths of another process. Besides distributional properties of the process, we present two applications to…
On weak convergence of quasi-infinitely divisible laws
本文研究了拟无穷可分律的弱收敛准则，在自然假设下将其弱收敛与 Lévy--Khinchine 谱函数的一种特殊收敛联系起来，补充了 Lindner、Pan 和 Sato（2018）的已知结果。
A Cluster Limit Theorem for Infinitely Divisible Point Processes
本文考虑一个点过程序列$(N_n)_{n \geq 1}$，它的点位于$\bR \verb2\2 \{0\}$的子集$E$中，并且满足渐近独立条件.我们的主要结果给出了$(N_n)_{n \geq 1}$依分布收敛于无穷可整除点过程$N$的一些充要条件。作为应用，我们讨论了基于正则变化序列的离散过程和点过程。
Let $Φ$ be a nuclear space and let $Φ'_β$ denote its strong dual. In this work we establish the one-to-one correspondence between infinitely divisible measures on $Φ'_β$ and Lévy processes taking values in $Φ'_β$. Moreover, we prove the Lévy-Itô decomposition, the Lévy-Khintchine formula and the existence of càdlàg versions for $Φ'_β$-valued Lévy processes. A characterization for Lévy measures on $Φ'_β$ is also established. Finally, we prove the Lévy-Khintchine formula for infinitely divisible measures on $Φ'_β$.
This paper studies the invertibility property of continuous time moving average processes driven by a Lévy process. We provide of sufficient conditions for the recovery of the driving noise. Our assumptions are specified via the kernel involved and the characteristic triplet of the background driving Lévy process.
A Generalised Gangolli-Levy-Khintchine Formula for Infinitely Divisible Measures and Levy Processes on Semi-Simple Lie Groups and Symmetric Spaces
在1964年R.Gangolli出版了列维-欣钦型公式，其特点是$K$双不变无限可分概率测度的对称空间$G/K$。他的主要工具是哈里什-钱德拉的球面函数，他用来构建一个概括的傅里叶变换的措施。在本文中，我们使用广义球函数（或爱森斯坦积分）和扩展这些，我们构造使用表示论，以获得这样的表征任意无限可分的概率测度上的非紧对称空间。我们对双曲空间的例子作了一些详细的考虑。
An infinitely divisible distribution on $\mathbb{R}$ is a probability measure $μ$ such that the characteristic function $\hatμ$ has a Lévy-Khintchine representation with characteristic triplet $(a,γ, ν)$, where $ν$ is a Lévy measure, $γ\in\mathbb{R}$ and $a\ge 0$. A natural extension of such distributions are quasi-infinitely distributions. Instead of a Lévy measure, we assume that $ν$ is a "signed Lévy measure", for further information on the definition see [\ref{Lindner}]. We show that a distribution $μ=pδ_{x_0}+(1-p)μ_{ac}$ with $p>0$ and $x_0 \in \mathbb{R}$, where $μ_{ac}$ is the absolute…
We prove a complete class theorem that characterizes \emph{all} stationary time reversible Markov processes whose finite dimensional marginal distributions (of all orders) are infinitely divisible. Aside from two degenerate cases (iid and constant), in both discrete and continuous time every such process with full support is a branching process with Poisson or Negative Binomial marginal univariate distributions and a specific bivariate distribution at pairs of times. As a corollary, we prove that every nondegenerate stationary integer valued processes constructed by the Markov thinning process…
The limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions
证明了$R^d$上的五类无穷可分分布（Jurek类、Goldie-Steutel-Bondesson类、自分解分布类、Thorin类和广义型$G$分布类）的嵌套子类的极限与稳定分布类的闭包相同.还给出了更一般的结果。
A short memory condition for infinitely divisible random fields
本文为无穷可分移动平均随机场提出短程依赖的充分条件，在对称稳定情形下与已有条件一致。
An essential character for a distribution to play a central role in the limit theory is infinite divisibility. In this note, we prove that the Conway-Maxwell-Poisson (CMP) distribution is infinitely divisible iff it is the Poisson or geometric distribution. This explains that, despite its applications in a wide range of fields, there is no theoretical foundation for the CMP distribution to be a natural candidate for the law of small numbers.
Concentration for Infinitely Divisible Vectors with Independent Components
对于各类Lipschitz函数，给出了具有独立分量和有限指数矩的无穷可分随机向量的无量纲集中不等式.
Limit Theory for the Sample Autocovariance for Heavy Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows
本文研究了一类平稳无穷可分过程的样本自协方差和样本自相关的极限定理。我们考虑的情况下，无限可分的过程中有重尾边缘和产生的保守流。有趣的是，样本自协方差的增长率不仅取决于边缘的重尾性，而且还取决于过程的记忆长度。虽然这一特点首先观察到的\cite{resnick:samorodnitsky:xue:2000}为一些非常具体的过程，我们将提出一个更普遍的框架，从无限遍历论的观点。因此，可以更全面地讨论样本自协方差的渐近性。
In this paper we analyze fractional Fokker-Planck equation describing subdiffusion in the general infinitely divisible (ID) setting. We show that in the case of space-time-dependent drift and diffusion and time-dependent jump coefficient, the corresponding stochastic process can be obtained by subordinating two-dimensional system of Langevin equations driven by appropriate Brownian and Levy noises. Our result solves the problem of stochastic representation of subdiffusive Fokker-Planck dynamics in full generality.
Nonnormal small jump approximation of infinitely divisible distributions
我们考虑一类非正态逼近的无穷可分分布，它包含复合泊松，伽玛，和正态分布。近似依赖于实现更高阶的累积量匹配，以获得更高的近似误差衰减率。近似的参数很容易确定。在许多情况下，近似分布的随机抽样的计算复杂性与正态近似具有相同的数量级。推导了以总方差距离表示的误差界。单变量和多变量的情况下的近似被认为是。
In this article the relation between the tail behaviours of a free regular infinitely divisible (positively supported) probability measure and its Lévy measure is studied. An important example of such a measure is the compound free Poisson distribution, which often occurs as a limiting spectral distribution of certain sequences of random matrices. We also describe a connection between an analogous classical result of Embrechts et al. [1979] and our result using the Bercovici-Pata bijection.
We prove a functional central limit theorem for partial sums of symmetric stationary long range dependent heavy tailed infinitely divisible processes with a certain type of negative dependence. Previously only positive dependence could be treated. The negative dependence involves cancellations of the Gaussian second order. This leads to new types of limiting processes involving stable random measures, due to heavy tails, Mittag-Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations.
Lévy Processes on $U_q(g)$ as Infinitely Divisible Representations
双代数上的Lévy过程是无限可分表示的族。我们在量子代数$U_q(g)$的紧致形式上对Lévy过程的生成元进行了分类，其中$g$是一个简单的李代数.然后，我们展示了如何从它们的生成器重建过程本身，并研究了几个经典的随机过程，可以与这些过程。
In this work we present different results concerning mixing properties of multivariate infinitely divisible (ID) stationary random fields. First, we derive some necessary and sufficient conditions for mixing of stationary ID multivariate random fields in terms of their spectral representation. Second, we prove that (linear combinations of independent) mixed moving average fields are mixing. Further, using a simple modification of the proofs of our results we are able to obtain weak mixing versions of our results. Finally, we prove the equivalence of ergodicity and weak mixing for multivariate …
Classes of infinitely divisible distributions on R^d related to the class of selfdecomposable distributions
本文研究R^d上新的一类无穷可分分布.首先，重新讨论了Jurek类和自分解分布类之间的具有连续参数的连接类。其次，扩展了参数的取值范围，构造了新的类，并给出了关于Levy过程的随机积分的刻画。最后，讨论了这些类的嵌套子类，并用随机整表示和Levy测度两种方法刻画了它们。
We completely characterize $Δ$- and local subexponentialities of positive-half compound Poisson distributions and extend the characterization on two-sided distributions. Moreover, $Δ$-subexponentiality of infinitely divisible distributions is characterized with new conditions, and local subexponentiality is newly characterized in the two-sided case. In the process closedness properties of these subexponentialities are derived, particularly for distributions on $\R$. Most results are obtained by exploiting monotonic-type assumptions. We apply our results to distributions of supremum of a random…
Generalized Coverage Processes with Infinitely Divisible Finite Dimensional Distributions
本文定义了一类具有无穷可分有限维分布和特定相关结构的广义覆盖过程，该结构可视为经典奥本海姆-乌尔申贝克过程的推广，包括M/GI/∞过程。通过建立适当的极限定理，展示了此类过程作为独立ON/OFF马尔可夫过程叠加的极限自然出现。
We consider questions related to the well-known conjecture due to Embrechts and Goldie on the closedness of different classes of heavy- and light-tailed distributions with respect to convolution roots. We show that the class L(γ)\cap OS is not closed under convolution roots related to an infinitely divisible distribution for any γ\ge0, i.e. we provide examples of infinitely divisible distributions belonging to this class such that the corresponding Levy spectral distribution does not. We also prove a similar statement for the class (L(γ)\cap OS)\ S(γ). In order to facilitate our analysis, we e…
Distributional comparison for non-commutative infinitely divisible probability measures
针对布尔、经典和自由卷积下的无穷可分分布，利用累积量差异估计非交换Wasserstein距离的上界。
The hitting partitions are random partitions that arise from the investigation of so-called hitting scenarios of max-infinitely-divisible (max-i.d.)~distributions. We study a class of max-i.d.~laws with exchangeable hitting partitions obtained by size-biased sampling from the jumps of a Lévy subordinator. We obtain explicit formulae for the distributions of these partitions in the case of the multivariate $α$-logistic and another family of exchangeable max-i.d.\ distributions. Specifically, the hitting partitions for these two cases are shown to coincide with the well-known Poisson--Dirichlet …
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 me…
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