Infinite-dimensional random fields on semigroups. (English. Russian original) Zbl 0591.60006

Ponomarenko, A. I.

Theory Probab. Math. Stat. 30, 153-158 (1985); translation from Teor. Veroyatn. Mat. Stat. 30, 136-142 (1984).

Let X be a complex locally convex space, X’ its topological conjugate, \(H=L_ 2(\Omega,{\mathcal A},P)\) the Hilbert space of complex random variables of second order and \({\mathcal L}(X',H)\) the space of linear continuous operators A:X’\(\to H\). In this paper families of operators \(A_ t\in {\mathcal L}(X',H)\), \(t\in T\), which are called generalized random functions of second order with values in X, are considered. In the case when T is a semigroup such a family is called a generalized random field. In previous papers [On the spectral theory of infinite-dimensional homogeneous in the broad sense random fields on groups. Visnik Kiiv. Univ. 1969, No.11, Ser. Mat. Mekh. 114-121 (1969) and Teor. Veroyatn. Mat. Stat. 7, 110-121 (1972; Zbl 0253.60046)] the author considered the case when X is a Hilbert or Banach space.

Reviewer: V.Paulauskas

MSC:

60B11	Probability theory on linear topological spaces
60G60	Random fields
20M30	Representation of semigroups; actions of semigroups on sets
60B15	Probability measures on groups or semigroups, Fourier transforms, factorization

Keywords:

locally convex space; generalized random functions; generalized random field

Citations:

Zbl 0253.60046