Random Measures and Applications

Abstract

A mapping Z(·) from a δ-ring ℬ0(ℝ) into the vector space of random variables L p (P) is a vector-valued measure if it is σ-additive in the metric of its range. It is a vector measure if the range is a Banach space and a random measure if also its values are independent on disjoint sets. An important reason for this study is to construct integrals relative to such Zs, which typically do not have finite variation. For this, it is essential to find a controlling (σ-finite) measure for Z that is not available if 0 <p < 1, and here the random measure is taken to be p-stable and utilize properties of infinitely divisible distributions. In the case of p = 2, Z(·) induces a bimeasure, and if p > 2 is an integer it induces a polymeasure, either of which need not be (signed) measures on product spaces. Important applications lead to all these possibilities. In all those cases, a detailed analysis of vector-valued set functions is presented, with special focus for the cases of 0 <p < 1 and p = 2 where probability and Bochner's L 2, 2 boundedness plays a key role. Specialization if Z is stationary, harmonizable, and/or isotropic are discussed using the group structure of ℝ n , n ≥ 1, extending it for an lca group G. If Z is Banach valued or a quasi-martingale measure, methods of obtaining integrals are outlined in the last section, and open problems motivated by applications are pointed out at various places.