Gaussian random measures
Let (S,\({\mathcal S})\) be a measurable space. A Gaussian process \(\eta =\{\eta (A):\quad A\in {\mathcal S}\}\) is a Gaussian random measure if for each \(\omega\), \(A\to \eta (\omega,A)\) is a signed measure on \({\mathcal S}\). (Note well that this is stronger than the definition of a second-order random measure, which only need satisfy \(\xi (\cup^{\infty}_{i=1}A_ i)=\sum^{\infty}_{i=1}\xi (A_ i)\) in the sense of \(L_ 2\)- convergence whenever the \(A_ i\) are disjoint.) Given a Gaussian random measure \(\eta\) with \(E[\eta (A)]=0\), \(A\in {\mathcal S}\), its covariance kernel is the set function on \({\mathcal S}\times {\mathcal S}\) defined by \(\nu_{\eta}(A,B)=E[\eta (A)\eta (B)],\) and is a signed bimeasure, i.e., is countably additive in either variable with the other held fixed. It is shown that a finite, symmetric signed bimeasure \(\nu_ 0\) is the covariance kernel of some Gaussian random measure if and only if (i) \(\nu_ 0\) is positive definite: \(\sum^{n}_{i,j=1}a_ ia_ j\nu_ 0(A_ i,A_ j)\geq 0\) for all choices of n, \(a_ 1,...,a_ n\) and \(A_ 1,...,A_ n\); and (ii) there is a constant c such that \(\sum_{i}\nu_ 0(B_ i,B_ i)^{1/2}\leq c\) for every countable \({\mathcal S}\)-partition \((B_ i)\) of S. It is further shown that every Gaussian random measure has the property that \(E[\exp \{\alpha | \eta |^ 2(S)\}]<\infty\) for \(\alpha\) sufficiently small, where \(| \eta |\) is the total variation measure of \(\eta\) (defined \(\omega\)-wise), and admits a representation \(\eta (A)=\int_{A}p(y)\mu (dy),\) where \(\{p(y):\quad y\in {\mathcal S}\}\) is a Gaussian process and \(\mu (A)=E[| \eta | (A)]\) (thus \(\mu\) is deterministic).
scientific article; zbMATH DE number 3960682