Finite version of the $q$-analogue of de Finetti's theorem

TL;DR

This paper develops an asymptotic version of the $q$-analogue of de Finetti's theorem, demonstrating that the convergence rate of $q$-exchangeable measures is of order $q^n$, thus extending classical probabilistic symmetry results.

Contribution

It introduces an asymptotic formulation of the $q$-de Finetti theorem and determines the optimal convergence rate as $q^n$, advancing understanding of $q$-exchangeability.

Findings

Convergence rate of order $q^n$ for $q$-exchangeable measures.

Uses convex structure of $q$-exchangeable probability measures.

Provides an asymptotic version of the $q$-analogue of de Finetti's theorem.

Abstract

Let $q \in (0,1)$. We formulate an asymptotic version of the $q$-analogue of de Finetti's theorem. Using the convex structure of the space of $q$-exchangeable probability measures, we show that the optimal rate of convergence is of order $q^n$.