For a stationary moving average random field, a non-parametric low frequency estimator of the Lévy density of its infinitely divisible independently scattered integrator measure is given. The plug-in estimate is based on the solution $w$ of the linear integral equation $v(x) = \int_{\mathbb{R}^d} g(s) w(h(s)x)ds$, where $g,h:\mathbb{R}^d \rightarrow \mathbb{R}$ are given measurable functions and $v$ is a (weighted) $L^2$-function on $\mathbb{R}$. We investigate conditions for the existence and uniqueness of this solution and give $L^2$-error bounds for the resulting estimates. An application to pure jump moving averages and a simulation study round off the paper.
The object of this paper is to study the asymptotic dependence structure of the linear time series models with infinitely divisible innovations by the use of their characteristic functions. Autoregressive moving-average (ARMA) models and fractional autoregressive integrated moving-average (FARIMA) models are analyzed. As examples of infinitely divisible innovations, the class of radially absolute continuous distributions and general non-symmetric stable distributions are considered. The finite dimensional distributionsn of these models are also obtained.
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…
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.
Given a low frequency sample of an infinitely divisible moving average random field $\{\int_{\mathbb{R}^d} f(x-t)Λ(dx); \ t \in \mathbb{R}^d \}$ with a known simple function $f$, we study the problem of nonparametric estimation of the Lévy characteristics of the independently scattered random measure $Λ$. We provide three methods, a simple plug-in approach, a method based on Fourier transforms and an approach involving decompositions with respect to $L^2$-orthonormal bases, which allow to estimate the Lévy density of $Λ$. For these methods, the bounds for the $L^2$-error are given. Their numer…
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…
Ramachandran (1969, Theorem 8) has shown that for any univariate infinitely divisible distribution and any positive real number $α$, an absolute moment of order $α$ relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding L$\acute{\rm e}$vy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, is given by Sato (1999, Theorem 25.3). We extend Ramachandran's theorem to the multivariate case, keeping in mind the immediate requir…
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.
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.
The paper develops new methods of non-parametric estimation a compound Poisson distribution. Such a problem arise, in particular, in the inference of a Levy process recorded at equidistant time intervals. Our key estimator is based on series decomposition of functionals of a measure and relies on the steepest descent technique recently developed in variational analysis of measures. Simulation studies demonstrate applicability domain of our methods and how they positively compare and complement the existing techniques. They are particularly suited for discrete compounding distributions, not nec…
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.
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…
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 …
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…
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…
Let $σ,t\in{\mathbb{R}}$, $s=σ+\mathrm{i}t$, $Γ(s)$ be the Gamma function, $ζ(s)$ be the Riemann zeta function and $ξ(s):=s(s-1)π^{-s/2}Γ(s/2)ζ(s)$ be the complete Riemann zeta function. We show that $Ξ_σ(t):=ξ(σ-\mathrm{i}t)/ξ(σ)$ is a characteristic function for any $σ\in{\mathbb{R}}$ by giving the probability density function. Next we prove that the Riemann hypothesis is true if and only if each $Ξ_σ(t)$ is a pretended-infinitely divisible characteristic function, which is defined in this paper, for each $1/2<σ<1$. Moreover, we show that $Ξ_σ(t)$ is a pretended-infinitely divisible characte…
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…
