Let Pnβ and Qnβ 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 Mnβ be a random variable representing a 'macrostate' or
'global observable' of that system. We provide sufficient conditions, based on
the Radon-Nikodym derivative of Pnβ and Qnβ, for the set of typical values
of Mnβ obtained relative to Pnβ to be the same as the set of typical values
obtained relative to Qnβ in the limit nββ. 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.Comment: v1: 16 pages. v2: 17 pages, precisions, examples and references
added. v3: Minor typos corrected. Close to published versio