Asymptotic equivalence of probability measures and stochastic processes

TL;DR

This paper establishes conditions under which two probabilistic models become indistinguishable in their predictions of macroscopic properties as the system size grows large, extending classical thermodynamic ensemble equivalence.

Contribution

It generalizes the concept of ensemble equivalence to broader probability measures and stochastic processes using Radon-Nikodym derivatives.

Findings

Sets of typical macrostate values coincide asymptotically

Extends thermodynamic-limit ensemble equivalence to general measures

Provides sufficient conditions for measure asymptotic equivalence

Abstract

Let $P_n$ and $Q_n$ be two probability measures representing two different probabilistic models of some system (e.g., an $n$-particle equilibrium system, a set of random graphs with $n$ vertices, or a stochastic process evolving over a time $n$) and let $M_n$ be a random variable representing a 'macrostate' or 'global observable' of that system. We provide sufficient conditions, based on the Radon-Nikodym derivative of $P_n$ and $Q_n$, for the set of typical values of $M_n$ obtained relative to $P_n$ to be the same as the set of typical values obtained relative to $Q_n$ in the limit $n\rightarrow\infty$. This extends to general probability measures and stochastic processes the well-known thermodynamic-limit equivalence of the microcanonical and canonical ensembles, related mathematically to the asymptotic equivalence of conditional and exponentially-tilted measures. In this more general sense, two probability measures that are asymptotically equivalent predict the same typical or macroscopic properties of the system they are meant to model.