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).     

Characterization of infinitely divisible multivariate gamma distributions

A particular class of p-dimensional exponential distributions have Laplace transforms |I + VT|^?1, V positive definite or positive semi-definite and T = diagonal (t₁,…, t_p). A characterization is given of when these Laplace transforms are infinitely divisible.     

Integrals with respect to infinitely divisible distributions in a Hilbert space

A connection between the existence of integrals with respect to infinitely divisible probability distributions in a Hilbert space and the existence of integrals with respect to their spectral measures of functions of certain classes is established.Translated from Matematicheskie Zametki, Vol. 11, No. 6, pp. 669–676, June, 1972.     

Characterization of palm distributions and infinitely divisible random measures

The purpose of the present paper is to characterize Campbell measures and Palm distributions of random measures and to apply these results in a new approach to the characterizations of infinitely divisible random measures by their Laplace functionals and their Palm distributions. The results on infinitely divisible random measures are well known. They can be found together with a detailed list of references in Kallenberg's monograph [2], which also contains proofs of almost all statements in Section 1 of this paper (see his note on page 9 concerning the Polish space setting).     

On multivariate infinitely divisible distributions

Simple conditions are given which characterize the generating function of a nonnegative multivariate infinitely divisible random vector. Necessary conditions on marginals, linear combinations, tail behavior, and zeroes are discussed, and a sufficient condition is given. The latter condition, which is a multivariate generalization of ordinary log-convexity, is shown to characterize only certain products of univariate infinitely divisible distributions.     

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.     

Approximation by infinitely divisible distributions in the multidimensional case

Let be the collection of parallelepipeds in R with edges parallel with the coordinate axes and let be the collection of closed sets in R. Let (G, H)=inf {G{A}H{A}+, H{A}G{A}+ for any; L(G, H)= inf {G{A}H{A}+, H{A}G{A}+ for any, where G, H are distributions in . In the paper one gives the proofs of results announced earlier by the author (Dokl. Akad. Nauk SSSR,253, No. 2, 277–279 (1980)). One considers the problem of the approximation of the distributions of sums of independent random vectors with the aid of infinitely divisible distributions. One obtains estimates for the distances (·, ·), L(·, ·) and. It is proved that, where 0p_i1, ; E is the distribution concentrated at zero; V_i(i=1, ..., n) are arbitrary distributions; the products and the exponentials are understood in the sense of convolution.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 89–103, 1983.     

Generalized renewal sequences and infinitely divisible lattice distributions

We introduce an increasing set of classes Γ_a (0?α?1) of infinitely divisible (i.d.) distributions on {0,1,2,…}, such that Γ₀ is the set of all compound-geometric distributions and Γ₁ the set of all compound-Poisson distributions, i.e. the set of all i.d. distributions on the non-negative integers. These classes are defined by recursion relations similar to those introduced by Katti [4] for Γ₁ and by Steutel [7] for Γ₀. These relations can be regarded as generalizations of those defining the so-called renewal sequences (cf. [5] and [2]). Several properties of i.d. distributions now appear as special cases of properties of the Γ_a'.     

On a characterization of infinitely divisible characteristic functionals on a Hubert space

Summary A characterization of infinitely divisible characteristic functional on a Hilbert space, analogous to that of Johansen [1], is given.     

On the closure under infinitely divisible distribution roots

Lithuanian Mathematical Journal - For some γ &gt; 0, we show that the distribution class (?(γ) ∩??S)\S(γ) is not closed under infinitely divisible...