Reversible Markov kernels and involutions on product spaces

TL;DR

This paper explores the relationship between independence-preserving involutions and reversible Markov kernels, introducing an involutive augmentation and demonstrating how certain kernels characterize geometric laws.

Contribution

It introduces an involutive augmentation framework linking IP involutions to reversible Markov kernels, with applications to characterizing geometric distributions.

Findings

Reversible Markov kernels relate to IP involutions.

Involutive augmentation H connects to f-generated kernels.

Random walk kernels with reflecting barriers characterize geometric laws.

Abstract

In this paper the relations between independence preserving (IP) involutions and reversible Markov kernels are investigated. We introduce an involutive augmentation H = (f, g_f) of a measurable function f and relate the IP property of H to f-generated reversible Markov kernels. Various examples appeared in the literature are presented as particular cases of the construction. In particular, we prove that the IP property generated by the (reversible) Markov kernel of random walk with a reflecting barrier at the origin characterizes geometric-type laws