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Exponentially equivalent measures

Definition[edit]

Let ${\displaystyle (M,d)}$ be a metric space and consider two one-parameter families of probability measures on ${\displaystyle M}$, say ${\displaystyle (\mu _{\varepsilon })_{\varepsilon >0}}$ and ${\displaystyle (\nu _{\varepsilon })_{\varepsilon >0}}$. These two families are said to be exponentially equivalent if there exist

• a one-parameter family of probability spaces ${\displaystyle (\Omega ,\Sigma _{\varepsilon },P_{\varepsilon })_{\varepsilon >0}}$,
• two families of ${\displaystyle M}$-valued random variables ${\displaystyle (Y_{\varepsilon })_{\varepsilon >0}}$ and ${\displaystyle (Z_{\varepsilon })_{\varepsilon >0}}$,

such that

• for each ${\displaystyle \varepsilon >0}$, the ${\displaystyle P_{\varepsilon }}$-law (i.e. the push-forward measure) of ${\displaystyle Y_{\varepsilon }}$is${\displaystyle \mu _{\varepsilon }}$, and the ${\displaystyle P_{\varepsilon }}$-law of ${\displaystyle Z_{\varepsilon }}$is${\displaystyle \nu _{\varepsilon }}$,
• for each ${\displaystyle \delta >0}$, "${\displaystyle Y_{\varepsilon }}$ and ${\displaystyle Z_{\varepsilon }}$ are further than ${\displaystyle \delta }$ apart" is a ${\displaystyle \Sigma _{\varepsilon }}$-measurable event, i.e.

${\displaystyle {\big \{}\omega \in \Omega {\big |}d(Y_{\varepsilon }(\omega ),Z_{\varepsilon }(\omega ))>\delta {\big \}}\in \Sigma _{\varepsilon },}$

• for each ${\displaystyle \delta >0}$,

${\displaystyle \limsup _{\varepsilon \downarrow 0}\,\varepsilon \log P_{\varepsilon }{\big (}d(Y_{\varepsilon },Z_{\varepsilon })>\delta {\big )}=-\infty .}$

The two families of random variables ${\displaystyle (Y_{\varepsilon })_{\varepsilon >0}}$ and ${\displaystyle (Z_{\varepsilon })_{\varepsilon >0}}$ 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 ${\displaystyle (\mu _{\varepsilon })_{\varepsilon >0}}$ with good rate function ${\displaystyle I}$, and ${\displaystyle (\mu _{\varepsilon })_{\varepsilon >0}}$ and ${\displaystyle (\nu _{\varepsilon })_{\varepsilon >0}}$ are exponentially equivalent, then the same large deviations principle holds for ${\displaystyle (\nu _{\varepsilon })_{\varepsilon >0}}$ with the same good rate function ${\displaystyle I}$.

References[edit]

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