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

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Publisher
Springer US
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
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

References

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