The theory of stochastic processes has been traditionally developed in terms of random variables and their joint distributions. This is not surprising since the definition of a stochastic process is abstracted from numerical statistical data that are empirically observed and to which the notion of a random variable is well suited. There is, however, another point of view for the study of a stochastic process. One may consider the family of all Boolean _-subalgebras of the sample space which are associated with the random variables of a stochastic process, and the combinatorial properties of such a family of _-subalgebras. Except in the cases of a stochastic process consisting of a set of independent random variables, and to a lesser extent the case of a martingale to which one associates an increasing or decreasing family of _-subalgebras this second point of view has been neglected, perhaps because of the measure-theoretic nitty-gritty that it demanded.
The objective of the present paper is that of pushing this second point of view one step further by singling out and studying a property of the families of _-subalgebras that are associated with certain stochastic processes, notably Gaussian processes. This property has been abstracted from previous knowledge of the very special case of a finite sample space. In this case, every _-subalgebra is associated with a partition of the underlying sample space. In several notable instances, families of partitions occur in which every pair of partitions commutes (cf. Section 5.1). The author was able to generalize the notion of commutativity of partitions to arbitrary Boolean algebras. In this paper, a further generalization is proposed, that of sigma-commuting Boolean _-algebras.
Let B and C be _-subalgebras of a sample space, and let D be their intersection. We say that the _-subalgebras B and C sigma-commute if for every pair of events b belonging to B and c belonging to C, we have P D (b & c)=0 if and only if P D (b) } P D (c)=0, where P D is the conditional probability relative to the _-subalgebra D.