Exponentially equivalent measures

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In mathematics, exponential equivalence of measures is how two sequences or families of probability measures are “the same” from the point of view of large deviations theory.

Definition[edit]

Let (Md) be a metric space and consider two one-parameter families of probability measures on M, say (με)ε>0 and (νε)ε>0. These two families are said to be exponentially equivalent if there exist

  • a one-parameter family of probability spaces ((Ω, ΣεPε))ε>0,
  • two families of M-valued random variables (Yε)ε>0 and (Zε)ε>0,

such that

  • for each ε > 0, the Pε-law (i.e. the push-forward measure) of Yε is με, and the Pε-law of Zε is νε,
  • for each δ > 0, “Yε and Zε are further than δ apart” is a Σε-measurable event, i.e.
\big\{ \omega \in \Omega \big| d(Y_{\varepsilon}(\omega), Z_{\varepsilon}(\omega)) > \delta \big\} \in \Sigma_{\varepsilon},
  • for each δ > 0,
\limsup_{\varepsilon \downarrow 0} \varepsilon \log \mathbf{P}_{\varepsilon} \big[ d(Y_{\varepsilon}, Z_{\varepsilon}) > \delta \big] = - \infty.

The two families of random variables (Yε)ε>0 and (Zε)ε>0 are also said to be exponentially equivalent.

Properties[edit]

The main use of exponential equivalence is that as far as large deviations principles are concerned, exponentially equivalent families of measures are indistinguishable. More precisely, if a large deviations principle holds for (με)ε>0 with good rate function I, and (με)ε>0 and (νε)ε>0 are exponentially equivalent, then the same large deviations principle holds for (νε)ε>0 with the same good rate function I.

References[edit]

  • Dembo, Amir; Zeitouni, Ofer (1998). Large deviations techniques and applications. Applications of Mathematics (New York) 38 (Second edition ed.). New York: Springer-Verlag. pp. xvi+396. ISBN 0-387-98406-2. MR 1619036.  (See section 4.2.2)