Exponentially equivalent measures

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.

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 ${\displaystyle I}$.

• 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)