On a certain class of infinitely divisible distributions

We characterize the class of distribution functions Φ(x), which are limits in the following sense: there exist a sequence of independent and equally distributed random variables {ξ _n }, numerical sequences {a _k }, {b _k } and natural numbers {n _k } such that $$\mathop {lim}\limits_{k \to \infty } Prob\left\{ {\frac{1}{{a_k }}\mathop {\Sigma }\limits_{k = 1}^{n_k } \xi _k - b_k< x} \right\} = \Phi (x)$$ and $$\mathop {\lim \inf }\limits_{k \to \infty } (n_k /n_{k + 1} ) > 0$$ .     

The asymptotics of multidimensional infinitely divisible distributions

The present article deals with the asymptotics at infinity of multidimensional infinitely divisible distributions with the support in a cone. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russian, 1995, Part III.     

Characterization of a class of infinitely divisible distributions in Hilbert space

It is established that the spectral measure of an infinitely divisible distribution F in a Hilbert space H is concentrated in a sphere of finite radius if and only if the integral ∫ _H exp (α∥x∥ In (∥x∥+1))dF is finite for some numberα>0. If this integral is finite for anyα>0 then the infinitely divisible distribution F is normal (maybe, degenerate).     

A class of infinitely divisible distributions connected to branching processes and random walks

A class of infinitely divisible distributions on {0,1,2,…} is defined by requiring the (discrete) Lévy function to be equal to the probability function except for a very simple factor. These distributions turn out to be special cases of the total offspring distributions in (sub)critical branching processes and can also be interpreted as first passage times in certain random walks. There are connections with Lambert's W function and generalized negative binomial convolutions.     

Characterization of extendible distributions with exponential minima via processes that are infinitely divisible with respect to time

We present a stochastic representation for multivariate extendible distributions with exponential minima (exEM), whose components are conditionally iid in the sense of de Finetti’s theorem. It is shown that the “exponential minima property” is in one-to-one correspondence with the conditional cumulative hazard rate process being infinitely divisible with respect to time (IDT). The Laplace exponents of non-decreasing IDT processes are given in terms of a Bernstein function applied to the state space variable and are linear in time. Examples for IDT processes comprise killed Lévy subordinators, monomials whose slope is randomized by a stable random variable, and several combinations thereof. As a byproduct of our results, we provide an alternative proof (and a mild generalization) of the important conclusion in Genest and Rivest (Stat. Probab. Lett. 8:207211, 1989), stating that the only copula which is both Archimedean and of extreme-value kind is the Gumbel copula. Finally, we show that when the subfamily of strong IDT processes is used in the construction leading to exEM, the result is the proper subclass of extendible min-stable multivariate exponential (exMSMVE) distributions.     

On the decomposition of infinitely divisible characteristic functions of several variables

Summary We prove here some theorems describing infinitely divisible characteristic functions defined on R ⁿ which have positive Poisson spectrum and belong to the class I ₀ of characteristic functions without indecomposable factor. These theorems are generalizations to the case of several variables of results due to I.V. Ostrovskiy in the case of one variable.This work was supported by the National Science Foundation under grant NSF-GP-6175.     

On infinitely divisible distributions with light tails of Lévy measures

We show that if the tail of a Lévy measure is light, then the same holds for the tail of the corresponding infinitely divisible distribution.     

Stability of decompositions of lattice infinitely divisible distributions of Linnik's class\mathfrak{L}

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 84–107, 1986.     

Stable convergence of the log-likelihood ratio to a mixture of infinitely divisible distributions

Some results concerning the asymptotic behavior of the log-likelihood ratio (LLR) and also of certain other random variables closely associated with the likelihood ratio are presented. More specifically, in the present paper we formulate the conditions for the stable convergence in distribution of the LLR for two sequences of the probability measures to a mixture of infinitely divisible distributions with finite variance. Moreover, the notion of a locally asymptotically mixed infinitely divisible (LAMID) sequence of parametric families of the probability measures is introduced, and it is shown that when a certain kind of differentiability-type regularity condition is satisfied, the given sequence of families satisfies the LAMID condition. These results extend and supplement the previous investigations of the author concerning non-Gaussian asymptotic distributions in statistics. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III.     

A class of infinitely divisible variance functions with an application to the polynomial case

Let be a natural exponential family on ??? with variance function (V, Ω). Here, Ω is the mean domain of and V is its variance expressed in terms of the mean μ ε Ω. In this note we prove the following result. Consider an open interval Ω = (0, b), 0 < b ∞, and a positive real analytic function V on Ω. If V² is absolutely monotone on [0, b) and V has the form μt(μ), where 1 and t is real analytic in a neighborhood of zero, then there exits an infinitely divisible natural exponential family with variance function (V, Ω). We illustrate this result with several examples of general nature.     

Stability estimates for the expansions of infinitely divisible distribution functions whose Gaussian components are of Linnik class ℭ. 2

Translated fromProblemy Ustoichivosti Stokhasticheskikh Modelei. Trudy Seminara, 1988, pp. 142–161.