Strongly harmonizable approximations of bounded continuous random fields
Let \(X: {\mathbb{R}}^ k\to L^ 2_{{\mathbb{C}}}(S,{\mathcal F},P)\) be a random field on the probability space (S,\({\mathcal F},P)\). X is said to be weakly harmonizable if there exists a \(\sigma\)-additive set function \(\mu\) : \({\mathcal B}({\mathbb{R}}^ k)\to L^ 2_{{\mathbb{C}}}(S,{\mathcal F},P)\) such that \[ X_ t=\int \exp (i<t,x>)\mu (dx)\quad (t\in {\mathbb{R}}^ k). \] If, in particular, there exists a complex measure M on \({\mathbb{R}}^ k\times {\mathbb{R}}^ k\) such that E[\(\mu\) (A)\(\overline{\mu (B)}]=M(A\times B)\) (A,B\(\in {\mathcal B}({\mathbb{R}}^ k))\), X is said to be strongly harmonizable. The authors show that each continuous and bounded random field X on \({\mathbb{R}}^ k\) is the limit, uniformly on compact subsets of \({\mathbb{R}}^ k\), of a sequence of strongly harmonizable random fields, each of them admitting an absolutely continuous spectral measure. This extends a result of \textit{H. Niemi} in Ann. Acad. Sci. Fenn., Ser. A I, Diss. 591 (1975; Zbl 0307.60034).
scientific article; zbMATH DE number 3988384