# Asymptotic Equivalence of Probability Measures and Stochastic Processes

Asymptotic Equivalence of Probability Measures and Stochastic Processes Let $$P_n$$ P n and $$Q_n$$ 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 $$M_n$$ 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 $$P_n$$ P n and $$Q_n$$ Q n , for the set of typical values of $$M_n$$ M n obtained relative to $$P_n$$ P n to be the same as the set of typical values obtained relative to $$Q_n$$ Q n in the limit $$n\rightarrow \infty$$ n → ∞ . 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Statistical Physics Springer Journals

# Asymptotic Equivalence of Probability Measures and Stochastic Processes

, Volume 170 (5) – Feb 1, 2018
17 pages

/lp/springer_journal/asymptotic-equivalence-of-probability-measures-and-stochastic-TQkvHEAj9h
Publisher
Springer Journals
Subject
Physics; Statistical Physics and Dynamical Systems; Theoretical, Mathematical and Computational Physics; Physical Chemistry; Quantum Physics
ISSN
0022-4715
eISSN
1572-9613
D.O.I.
10.1007/s10955-018-1965-5
Publisher site
See Article on Publisher Site

### Abstract

Let $$P_n$$ P n and $$Q_n$$ 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 $$M_n$$ 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 $$P_n$$ P n and $$Q_n$$ Q n , for the set of typical values of $$M_n$$ M n obtained relative to $$P_n$$ P n to be the same as the set of typical values obtained relative to $$Q_n$$ Q n in the limit $$n\rightarrow \infty$$ n → ∞ . 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.

### Journal

Journal of Statistical PhysicsSpringer Journals

Published: Feb 1, 2018

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