Stochastic Mappings and Random Distribution Fields. A Correlation Approach

TL;DR

This paper explores the structure of multivariate second order stochastic mappings and their covariance functions, extending classical stochastic process theory to include random distribution fields and measures with a unified correlation-based approach.

Contribution

It introduces a unified framework for multivariate second order stochastic mappings using reproducing kernel structures, extending classical theory to random distribution fields and measures.

Findings

Characterization of operator covariance functions

Extension of Wold type decomposition

Unified correlation-based framework

Abstract

This paper contains a study of multivariate second order stochastic mappings indexed by an abstract set $\Lambda$ in close connection to their operator covariance functions. The characterizations of the normal Hilbert module or of Hilbert spaces associated to such a multivariate second order stochastic mapping in terms of reproducing kernel structures are given, aiming not only to gather into a unified way some concepts from the field, but also to indicate an instrument for extending the very well elaborated theory of multivariate second order stochastic processes (or random fields) to the case of multivariate second order random distribution fields, including multivariate second order stochastic measures. In particular a general Wold type decomposition is extended and discussed in our framework.