Algebraic models for probability measures associated with stochastic processes

Abstract

Abstract. This paper initiates the study of probability measures corresponding to stochastic processes based on the Dinculeanu-Foias notion of algebraic models for probability measures. The main result is a general extension theorem of Kolmogorov type which can be summarized as follows: Let {(X, si¡, ¡i,), i e 1} be a directed family of probability measure spaces. Then there is an associated directed family of probability measure spaces {(G, &t, vt), i e 1} and a probability measure v on the »-algebra M generated by the 3)¡ such that (i) v(B) = v¡(B), Be &,, ie I, and (ii) for each iel the spaces (X, s/t, /it) and (G, &lt v¡) are conjugate. The importance of the main theorem is that under certain mild conditions there exists an embedding ip: X-* G such that the induced measures vt on G are extendable to v, although the measures n¡ on X may not be extendable. Using the algebraic model formulation, the Kolmogorov extension property and the notion of a representation of a directed family of probability measure spaces are discussed.1. Introduction. Let (í¿, sé, /x) be a probability measure space and T a subset of the real line. If (X, 3S) is some measurable space, then an jaZ-measurable function x(oS): Q -> X is called a random element in X. A random function, or stochastic process, defined on T with values in A" is a function x(t, tu): Tx £2 -> X such that x(t, of) is a random element in A" for each t e T. To fix ideas, we will restrict our attention to the classical case and take X= R (the real line) and 38 the a-algebra of Borel sets. Let S denote the set of all functions x = x(t) defined on T= [a, b], say.

Characterization and Duality of Projective and Direct Limits of Measures and Applications

“…A somewhat analogous method was considered by Dinculeanu and Foiaş [4] for a different purpose. Results of [23] based on [4] contain ideas similar to those of Sec. 3, but the present treatment is much more general.…”

Dynamical properties of endomorphisms, multiresolutions, similarity and orthogonality relations

“…Follows from Kolmogorov's inductive limit construction. For details, see [JT15, DJ14, GRPA10, DR08, DR07] and also [Hid80,Moh14,SSBR71].…”

“…Most of the arguments are already contained in the previous sections. Given (R, h, λ) as stated, the corresponding measure P on (Ω X , C ) is determined by (6.3) and Kolmogorov consistency [Hid80,Moh14,SSBR71]. And it then also follows from (6.3) that the two conditions (1a)-(1b) in the lemma are equivalent.…”

Algebraic models for probability measures on Banach spaces

“…These notions are due to Dinculeanu and Foias [2]. The study of algebraic models for probability measures associated with stochastic processes was initiated by Schreiber et al [8]. In virtually all areas of applied mathematics we encounter operator equations of the form (*) Tx == y where T:~~UJi, y is a UJi -valued random element; hence x is an~-valued random element.…”