Gaussian random measures

Abstract

A Gaussian random measure is a mean zero Gaussian process η( A), indexed by sets A in a σ-field, such that η( ΣA i )= Ση( A i ), where ΣA i indicates disjoint union and the series on the right is required to converge everywhere, so η is a random signed measure. (This is in contrast to so-called second order random measures, which only require quadratic mean convergence.) The covariance kernel of η is the signed bimeasure ν 0( A, B)= Eη( A) η( B). We give a characterization of those bimeasures which are covariance kernels of Gaussian random measures, and we show that every Gaussian random measure has an exponentially integrable total variation and is a.s. absolutely continuous with respect to a fixed finite measure on the state space.