{
  "id": "1708.02890",
  "version": "v1",
  "published": "2017-08-09T16:11:05.000Z",
  "updated": "2017-08-09T16:11:05.000Z",
  "title": "Asymptotic equivalence of probability measures and stochastic processes",
  "authors": [
    "Hugo Touchette"
  ],
  "comment": "16 pages",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "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 $n\\rightarrow\\infty$. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-09T16:11:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic process",
      "asymptotic equivalence",
      "well-known thermodynamic-limit equivalence",
      "general probability measures",
      "typical values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}