Markovianity and the Thompson Monoid $F^+$

TL;DR

This paper introduces a new invariance principle called partial spreadability, linking it to Markov properties and the Thompson monoid in noncommutative probability, and extends classical de Finetti theorems.

Contribution

It establishes a novel connection between partial spreadability, Markov sequences, and the Thompson monoid $F^+$ in noncommutative probability spaces.

Findings

Partial spreadability implies adaptation to a local Markov filtration.

Stationary Markov sequences can be represented via the Thompson monoid $F^+$.

A de Finetti theorem for stationary Markov sequences in classical probability.

Abstract

We introduce a new distributional invariance principle, called `partial spreadability', which emerges from the representation theory of the Thompson monoid $F^+$ in noncommutative probability spaces. We show that a partially spreadable sequence of noncommutative random variables is adapted to a local Markov filtration. Conversely we show that a large class of noncommutative stationary Markov sequences provides representations of the Thompson monoid $F^+$. In the particular case of a classical probability space, we arrive at a de Finetti theorem for stationary Markov sequences with values in a standard Borel space.