Stein Estimation for Infinitely Divisible Laws

On infinitely divisible distributions with polynomially decaying characteristic 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.

Why There Are No Gaps In The Support Of Non-Negative Integer-Valued Infinitely Divisible Laws?

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.

A characterization of ARMA and Fractional ARIMA models with infinitely divisible innovations

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.

A Supplement To The Bose-Dasgupta-Rubin (2002) Review Of Infinitely Divisible Laws And Processes

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…

On a linear functional for infinitely divisible moving average random fields

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…

A functional large and moderate deviation principle for infinitely divisible processes driven by null-recurrent markov chains

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 limiting behavior of some infinitely divisible exponential dispersion models

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…

An Inverse Problem for Infinitely Divisible Moving Average Random Fields

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…

A solution to a linear integral equation with an application to statistics of infinitely divisible moving averages

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 t…

几何无穷可分自回归模型

Geometric Infinitely Divisible Autoregressive Models

本文研究了几何无穷可分随机变量及其性质，提出了具有 gid 边缘分布的 AR 模型，并将其推广至 k 阶，同时给出了 AR(1) 过程的参数估计方法。

Moment properties of multivariate infinitely divisible laws and criteria for self-decomposability

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…

无穷可分位置模型中预测Kullback-Leibler遗憾的熵-能量恒等式

An Entropy-Energy Identity for Predictive Kullback-Leibler Regret in Infinitely Divisible Location Models

针对无穷可分位置模型，提出熵-能量恒等式将贝叶斯预测密度的遗憾表示为对称马尔可夫半群的Dirichlet形式能量，并给出可容许性的尾条件。

Nonparametric estimation of infinitely divisible distributions based on variational analysis on measures

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…

具有无限可分且厚尾先验测度的良好设定贝叶斯逆问题

Well-posed Bayesian Inverse Problems with Infinitely-Divisible and Heavy-Tailed Prior Measures

本文提出基于广义伽马分布的新的先验测度，用于p∈(0,1)的ℓp正则化技术，展示其厚尾、非凸和无限可分性质，并探讨无限可分先验测度的尾行为与莱维测度的联系，定义新的函数空间先验测度，解决贝叶斯逆问题的良定性。

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 …

A complete Riemann zeta distribution and the Riemann hypothesis

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…

无限可分隐私与超越 I：$s^2=2k$猜想的解决

Infinitely divisible privacy and beyond I: resolution of the $s^2=2k$ conjecture

本文提出了一种基于无限可分理论的微积分隐私框架，解决了$s^2=2k$猜想，并提供了非高斯极限和最优机制。