An orthogonal expansion of a homogeneous and isotropic random field on a Hilbert space. (English. Russian original) Zbl 0746.60055

Kuznetsova, O. M.

Theory Probab. Math. Stat. 42, 77-81 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 42, 67-72 (1990).

The author obtains an orthogonal expansion for a homogeneous isotropic mean-square continuous real-valued Gaussian random field \(\xi (t)\) on an infinite-dimensional Hilbert space. The main result is: there exist positive-definite kernels \(d_ m(x,y),\;m\geq 0\), on \(R_ +^ 2\) and orthogonal isotropic Gaussian fields \(\xi_ m(t),\;t\in H\), with covariance functions \(B_ m(t,s) = d_ m(r_ t,r_ s)(\alpha_ t,\alpha_ s)^ m\) such that \(\xi (t)=\sum_{m\geq 0}\xi_ m(t)\). (Here \(H\ni t = r_ t\alpha_ t\), where \(r_ t = \| t\|\) and \(\alpha_ t \in S_ \infty\), \(S_ \infty\) infinite-dimensional sphere).

Reviewer: Yu.S.Mishura (Kiev)

MSC:

60G60	Random fields
60B11	Probability theory on linear topological spaces

Keywords:

orthogonal expansion; Gaussian random field; Hilbert space