A transformation of gamma max-infinitely divisible laws viz. geometric gamma max-infinitely divisible laws is considered in this paper. Some of its distributional and divisibility properties are discussed and a random time changed extremal process corresponding to this distribution is presented. A new kind of invariance (stability) under geometric maxima is proved and a max-AR(1) model corresponding to it is also discussed.
Continuing the study reported in Satheesh (2001),(arXiv:math.PR/0304499 dated 01May2003) here we study certain aspects of randomization in infinitely divisible (ID) and max-infinitely divisible (MID) laws. They generalize ID and MID laws. In particular we study mixtures of ID & MID laws, its relation to random sums & random maximums, corresponding stationary processes & extremal processes and some of their properties. It is shown that mixtures of ID laws and mixtures of MID laws appear as limits of random sums and random maximums respectively. We identify a class of probability generating func…
Classes of multivariate and cone valued infinitely divisible Gamma distributions are introduced. Particular emphasis is put on the cone-valued case, due to the relevance of infinitely divisible distributions on the positive semi-definite matrices in applications. The cone-valued class of generalised Gamma convolutions is studied. In particular, a characterisation in terms of an Itô-Wiener integral with respect to an infinitely divisible random measure associated to the jumps of a Lévy process is established. A new example of an infinitely divisible positive definite Gamma random matrix is intr…
In this article, we give some reviews concerning negative probabilities model and quasi-infinitely divisible at the beginning. We next extend Feller's characterization of discrete infinitely divisible distributions to signed discrete infinitely divisible distributions, which are discrete pseudo compound Poisson (DPCP) distributions with connections to the Lévy-Wiener theorem. This is a special case of an open problem which is proposed by Sato(2014), Chaumont and Yor(2012). An analogous result involving characteristic functions is shown for signed integer-valued infinitely divisible distributio…
We show the equivalence of three properties for an infinitely divisible distribution: the subexponentiality of the density, the subexponentiality of the density of its Lévy measure and the tail equivalence between the density and its Lévy measure density, under monotonic-type assumptions on the Lévy measure density. The key assumption is that tail of the Lévy measure density is asymptotic to a non-increasing function or is eventually non-increasing. Our conditions are novel and cover a rather wide class of infinitely divisible distributions. Several significant properties for analyzing the sub…
We obtain dimension-free concentration inequalities for $\ell^p$-norms, $p\geq2$, of infinitely divisible random vectors with independent coordinates and finite exponential moments. Besides such norms, the methods and results extend to some other classes of Lipschitz functions.
We provide necessary and sufficient conditions on the characteristics of an infinitely divisible distribution under which its characteristic function $ϕ$ decays polynomially. Under a mild regularity condition this polynomial decay is equivalent to $1/ϕ$ being a Fourier multiplier on Besov spaces.
Remark.9 in Bose-Dasgupta-Rubin (2002) review states that when a non-negative integer-valued infinitely divisible law has an atom at unity then its support cannot have any gaps. Here one has two questions. (i) Why there are no gaps and (ii) Can there be gaps if the condition is not satisfied. Our investigation with these questions in mind centers on the implications of having and not having atoms at zero and unity. We give two examples/ constructions, which show that the remark needs modification and we modify it.
In this work infinitely divisible cylindrical probability measures on arbitrary Banach spaces are introduced. The class of infinitely divisible cylindrical probability measures is described in terms of their characteristics, a characterisation which is not known in general for infinitely divisible Radon measures on Banach spaces. Further properties of infinitely divisible cylindrical measures such as continuity are derived. Moreover, the result on the classification enables us to conclude new results on genuine Levy measures on Banach spaces.
This paper proves that if a discrete distribution is infinitely divisible (ID) with integer-valued components, then it has a mass at the origin, which also implies why certain ID discrete laws do not have gaps in its support. We argue that discrete laws also can be stable and such laws do have domain of attraction. Then we give certain recent developments and references not reported in the Bose Dasgupta Rubin (2002) review in Sankhya, and some examples in the topics; infinite divisibility and stability of discrete laws, random infinite divisibility, operator stable laws, class-L laws, Goldie-S…
We prove that a stationary max--infinitely divisible process is mixing (ergodic) iff its dependence function converges to 0 (is Cesaro summable to 0). These criteria are applied to some classes of max--infinitely divisible processes.
We consider random fields that can be represented as integrals of deterministic functions with respect to infinitely divisible random measures and show that these random fields are infinitely divisible.
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.
Investigates the size of the perturbation to the zero mean three dimensional Gaussian vector with infinitely divisible squares so that the infinite divisibility is retained.
Given a low-frequency sample of the infinitely divisible moving average random field $\{\int_{\mathbb{R}^d}f(t-x)Λ(dx), t\in \mathbb{R}^d\}$, in [13] we proposed an estimator $\hat{uv_0}$ for the function $\mathbb{R}\ni x\mapsto u(x)v_0(x)=(uv_0)(x)$, with $u(x)=x$ and $v_0$ being the Lévy density of the integrator random measure $Λ$. In this paper, we study asymptotic properties of the linear functional $L^2(\mathbb{R})\ni v\mapsto \left \langle v,\hat{uv_0}\right \rangle_{L^2(\mathbb{R})}$, if the (known) kernel function $f$ has a compact support. We provide conditions that ensure consistenc…
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.
Suppose $ E$ is a space with a null-recurrent Markov kernel $ P$. Furthermore, suppose there are infinite particles with variable weights on $ E$ performing a random walk following $ P$. Let $ X_{t}$ be a weighted functional of the position of particles at time $ t$. Under some conditions on the initial distribution of the particles the process $ (X_{t})$ is stationary over time. Non-Gaussian infinitely divisible (ID) distributions turn out to be natural candidates for the initial distribution and then the process $ (X_{t})$ is ID. We prove a functional large and moderate deviation principle f…
The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models considered in Benaych-Georges who introduced the models through their measures. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integra…
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…
Two methods to approximate infinitely divisible random fields are presented. The methods are based on approximating the kernel function in the spectral representation of such fields, leading to numerical integration of the respective integrals. Error bounds for the approximation error are derived and the approximations are used to simulate certain classes of infinitely divisible random fields.
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…
Consider an exponential dispersion model (EDM) generated by a probability $ μ$ on $[0,\infty )$ which is infinitely divisible with an unbounded Lévy measure $ν$. The Jorgensen set (i.e., the dispersion parameter space) is then $\mathbb{R}^{+}$, in which case the EDM is characterized by two parameters: $θ_{0}$ the natural parameter of the associated natural exponential family and the Jorgensen (or dispersion) parameter $t$. Denote by $EDM(θ_{0},t)$ the corresponding distribution and let $Y_{t}$ is a r.v. with distribution $EDM(θ_0,t)$. Then if $ν((x,\infty ))\sim -\ell \log x$ around zero we pr…
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.
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 …
The aim of this short note is to present the notion of IDT processes, which is a wide generalization of Lévy processes obtained from a modified infinitely divisible property. Special attention is put on a number of examples, in order to clarify how much the IDT processes either differ from, or resemble to, Lévy processes.
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.
This paper gives a complete characterization of infinitely divisible semimartingales, i.e., semimartingales whose finite dimensional distributions are infinitely divisible. An explicit and essentially unique decomposition of such semimartingales is obtained. A new approach, combining series decompositions of infinitely divisible processes with detailed analysis of their jumps, is presented. As an ilustration of the main result, the semimartingale property is explicitely determined for a large class of stationary increment processes and several examples of processes of interest are considered. …
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.
We investigate a Belinschi-Nica type semigroup for free and Boolean max-convolutions. We prove that this semigroup at time one connects limit theorems for freely and Boolean max-infinitely divisible distributions. Moreover, we also construct a max-analogue of Boolean-classical Bercovici-Pata bijection, establishing the equivalence of limit theorems for Boolean and classical max-infinitely divisible distributions.
We describe certain sufficient conditions for an infinitely divisible probability measure on a class of connected Lie groups to be embeddable in a continuous one-parameter convolution semigroup of probability measures. (Theorem 1.3). This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain Lie groups, including the so called Walnut group (Corollary 1.5). The embeddability is concluded also under certain other conditions (Corollary 1.4 and Theorem 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…
We give sharp estimate for the free energy of directed polymers in random environment in dimension 1+1. This estimate was known for a Gaussian environment, we extend it to the case where the law of the environment is infinitely divisible.
The article is devoted to stochastic processes with values in finite-dimensional vector spaces over infinite locally compact fields with non-trivial non-archimedean valuations. Infinitely divisible distributions are investigated. Theorems about their characteristic functionals are proved. Particular cases are demonstrated.
In this note we discuss the nature of gaps in the support of a discretely infinitely divisible distribution from the angle of compound Poisson laws/processes. The discussion is extended to infinitely divisible distributions on the non-negative real line.
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.
We show that, up to multiplication by constants, a Gaussian process has an infinitely divisible square if and only if its covariance is the Green function of a transient Markov process.
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.
We characterize the finite variation property for stationary increment mixed moving averages driven by infinitely divisible random measures. Such processes include fractional and moving average processes driven by Levy processes, and also their mixtures. We establish two types of zero-one laws for the finite variation property. We also consider some examples to illustrate our results.
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…
In this paper, we derive the bi-free analogue of the Lévy-Hinčin formula for compactly supported planar probability measures which are infinitely divisible with respect to the additive bi-free convolution introduced by Voiculescu. We also provide examples of bi-free infinitely divisible distributions with their bi-free Lévy-Hinčin representations. Furthermore, we construct the bi-free Lévy processes and the additive bi-free convolution semigroups generated by compactly supported planar probability measures.
In this article, we consider a sequence $(N_n)_{n \geq 1}$ of point processes, whose points lie in a subset $E$ of $\bR \verb2\2 \{0\}$, and satisfy an asymptotic independence condition. Our main result gives some necessary and sufficient conditions for the convergence in distribution of $(N_n)_{n \geq 1}$ to an infinitely divisible point process $N$. As applications, we discuss the exceedance processes and point processes based on regularly varying sequences.
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.
In 1964 R.Gangolli published a Lévy-Khintchine type formula which characterised $K$ bi-invariant infinitely divisible probability measures on a symmetric space $G/K$. His main tool was Harish-Chandra's spherical functions which he used to construct a generalisation of the Fourier transform of a measure. In this paper we use generalised spherical functions (or Eisenstein integrals) and extensions of these which we construct using representation theory to obtain such a characterisation for arbitrary infinitely divisible probability measures on a non-compact symmetric space. We consider the examp…
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…
The ever-growing appearance of infinitely divisible laws and related processes in various areas, such as physics, mathematical biology, finance and economics, has fuelled an increasing demand for numerical methods of sampling and sample path generation. In this survey, we review shot noise representation with a view towards sampling infinitely divisible laws and generating sample paths of related processes. In contrast to many conventional methods, the shot noise approach remains practical even in the multidimensional setting. We provide a brief introduction to shot noise representations of in…
