A module theoretic interpretation of multiplicity and rank of a stationary random process

This paper is in a sense a celebration of Fuhrmann’s book Linear Systems and Operators in Hilbert Space. Inspired by one of Fuhrmann’s unifying ideas, in this note we shall discuss module theory applied to stationary stochastic processes. In particular, we shall reframe the notion of multiplicity and rank of a second order stochastic process in a module theoretic framework. This will (hopefully) clarify some issues which are still occasion of some confusion in the literature.