$q$-exchangeability via quasi-invariance

TL;DR

This paper extends the concept of $q$-exchangeability to real-valued sequences using quasi-invariance, connecting it with Markov chains, random flags, and generalizing de Finetti's theorem for $q eq 1$.

Contribution

It introduces a new framework for $q$-exchangeability on real sequences, incorporating order and cocycles, and provides explicit constructions and connections to Markov processes and algebraic structures.

Findings

Extended $q$-analog of de Finetti's theorem to real-valued sequences

Constructed ergodic $q$-exchangeable measures via random shuffling

Linked $q$-exchangeability with Markov chains and invariant flags

Abstract

For positive $q\neq1$, the $q$-exchangeability of an infinite random word is introduced as quasi-invariance under permutations of letters, with a special cocycle which accounts for inversions in the word. This framework allows us to extend the $q$-analog of de Finetti's theorem for binary sequences---see Gnedin and Olshanski [Electron. J. Combin. 16 (2009) R78]---to general real-valued sequences. In contrast to the classical case of exchangeability ($q=1$), the order on $\mathbb{R}$ plays a significant role for the $q$-analogs. An explicit construction of ergodic $q$-exchangeable measures involves random shuffling of $\mathbb{N}=\{1,2,...\}$ by iteration of the geometric choice. Connections are established with transient Markov chains on $q$-Pascal pyramids and invariant random flags over the Galois fields.