Asymptotic Equivalence of Probability Measures and Stochastic Processes

Let and 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 be a random variable representing a "macrostate" or "global observable" of that system. We provide sufficient conditions, based on the Radon-Nikodym derivative of and , for the set of typical values of obtained relative to to be the same as the set of typical values obtained relative to in the limit . 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.