Exponentially equivalent measures
# Exponentially equivalent measures

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

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

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

such that

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

       ${\big \\omega \in \Omega {\big |}d(Y_{\varepsilon }(\omega ),Z_{\varepsilon }(\omega ))>\delta {\big \}}\in \Sigma _{\varepsilon },$
* for each $\delta >0$,

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

The two families of random variables $(Y_{\varepsilon })_{\varepsilon >0}$ and $(Z_{\varepsilon })_{\varepsilon >0}$ are also said to be exponentially equivalent.

## Properties

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 $(\mu _{\varepsilon })_{\varepsilon >0}$ with good rate function $I$, and $(\mu _{\varepsilon })_{\varepsilon >0}$ and $(\nu _{\varepsilon })_{\varepsilon >0}$ are exponentially equivalent, then the same large deviations principle holds for $(\nu _{\varepsilon })_{\varepsilon >0}$ with the same good rate function $I$.

