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D. A. Skorokhod

On Poisson and Gaussian measures in Euclidian spaces

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(A_k ) are independent ,

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

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 {A_k ∈ B(R^d ), k ≤ n} the random variables {μ(A_k ), k ≤ n} 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.

Random Operators and Stochastic Equations, Walter de Gruyter

Print ISSN: 0926-6364
Volume: 14, 03/2006
Pages: 23 - 34

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