Asymptotic Equivalence of Probability Measures and Stochastic Processes

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 (nrightarrow 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.