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Convergence of Tsirelson convolution systems of probability spaces

Remus Floricel, Patrick Melanson

Abstract

We associate two specific projective systems of probability spaces with any Tsirelson convolution system. If the projective limits of these systems exist, then we call the convolution system convergent and $K$-convergent, respectively. It is shown that convergent convolution systems give rise to continuous products of probability spaces, while $K$-convergent convolution systems lead to flow systems. We investigate the relationship between convergence and $K$-convergence, as well as their connections to two-parameter product systems of Hilbert spaces.

Convergence of Tsirelson convolution systems of probability spaces

Abstract

We associate two specific projective systems of probability spaces with any Tsirelson convolution system. If the projective limits of these systems exist, then we call the convolution system convergent and -convergent, respectively. It is shown that convergent convolution systems give rise to continuous products of probability spaces, while -convergent convolution systems lead to flow systems. We investigate the relationship between convergence and -convergence, as well as their connections to two-parameter product systems of Hilbert spaces.

Paper Structure

This paper contains 7 sections, 9 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Convolution systems of probability spaces and subproduct systems of Hilbert spaces
  4. Projective systems of probability spaces
  5. Finite Partitions
  6. Convergent convolution systems and associated CPPSs
  7. $K$-convergent convolution systems and associated flow systems

Key Result

Proposition \oldthetheorem

Let $\mathscr {S}=\{\Omega_{s,t}, \mathcal{F}_{s,t}, \mu_{s,t}, T_{r,s,t}\}_\mathbb{S}$ be a convolution system. Then the system $(\{(\Omega_I, \mathcal{F}_I, \mu_I)\}, \{T_{I,J}\})$ is a projective system of probability spaces over the directed set $(\mathcal{K}_{s,t},\subseteq)$, for all $s<t$ in

Theorems & Definitions (24)

  • Definition 2.1
  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition \oldthetheorem
  • proof
  • Definition 3.1
  • Theorem \oldthetheorem
  • proof
  • Definition 3.2
  • ...and 14 more