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Random Operators and Stochastic Equations

Editor-in-Chief: Girko, Vyacheslav

Managing Editor: Molchanov, S.

Editorial Board: Accardi, L. / Albeverio, Sergio / Carmona, R. / Casati, G. / Christopeit, N. / Domanski, C. / Drygas, Hilmar / Gupta, A.K. / Ibragimov, I. / Kirsch, Werner / Klein, A. / Kondratyev, Yuri / Kurotschka, V. / Leonenko, N. / Loubaton, Philippe / Orsingher, E. / Pastur, L. / Rodrigues, Waldyr A. / Shiryaev, Albert / Turbin, A.F. / Veretennikov, Alexandre

CiteScore 2018: 0.26

SCImago Journal Rank (SJR) 2018: 0.142
Source Normalized Impact per Paper (SNIP) 2018: 0.375

Mathematical Citation Quotient (MCQ) 2018: 0.11

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Volume 14, Issue 1


On Poisson and Gaussian measures in Euclidian spaces

D. A. Skorokhod

We consider random measures as measures with values in the Hilbert space of random variables on some probability space see [1,2].

A Poisson measure is the family of random variables satisfying the conditions

(P1) for any finite sequence for which the random variables π(Ak ) are independent ,

(P2) the random variable π(A) has a Poisson distribution with the parameter m(A) where m is a measure on B(Rd ) .

The main result of the article concerning Poisson measures is the proof of the existence of a sequence {x k(ω), k ≥ 1} of R d-valued random variables for which

A random measure is Gaussian if for any finite sequence {Ak B(Rd ), kn} the random variables {μ(Ak ), kn} have joint Gaussian distribution. As a rule measures were considered under the additional assumption that the values are independent for disjoint subsets.

We consider the existence of a measure in the general case.

Introduce the correlation function

It is proved that any positively defined function is the correlation function for some Gaussian measure.

About the article

Published in Print: 2006-03-01

Citation Information: Random Operators and Stochastic Equations rose, Volume 14, Issue 1, Pages 23–34, ISSN (Online) 1569-397x, ISSN (Print) 0926-6364, DOI: https://doi.org/10.1515/156939706776138002.

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