exponentially equivalent measures

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

• a one-parameter family of probability spaces $(\backslash Omega,\backslash Sigma\_\backslash varepsilon,P\_\backslash varepsilon)\_\{\backslash varepsilon\; >0\}$,
• two families of $M$-valued random variables $(Y\_\backslash varepsilon)\_\{\backslash varepsilon\; >0\}$ and $(Z\_\backslash varepsilon)\_\{\backslash varepsilon\; >0\}$,

such that

• for each $\backslash varepsilon\; >0$, the $P\_\backslash varepsilon$-law (i.e. the push-forward measure) of $Y\_\backslash varepsilon$ is $\backslash mu\_\backslash varepsilon$, and the $P\_\backslash varepsilon$-law of $Z\_\backslash varepsilon$ is $\backslash nu\_\backslash varepsilon$,
• for each $\backslash delta\; >0$, "$Y\_\backslash varepsilon$ and $Z\_\backslash varepsilon$ are further than $\backslash delta$ apart" is a $\backslash Sigma\_\backslash varepsilon$-measurable event, i.e.

::$\backslash big\backslash \{\; \backslash omega\; \backslash in\; \backslash Omega\; \backslash big|\; d(Y\_\{\backslash varepsilon\}(\backslash omega),\; Z\_\{\backslash varepsilon\}(\backslash omega))\; >\; \backslash delta\; \backslash big\backslash \}\; \backslash in\; \backslash Sigma\_\{\backslash varepsilon\},$

• for each $\backslash delta\; >0$,

::$\backslash limsup\_\{\backslash varepsilon\; \backslash downarrow\; 0\}\backslash ,\; \backslash varepsilon\; \backslash log\; P\_\backslash varepsilon\; \backslash big(\; d(Y\_\backslash varepsilon,\; Z\_\backslash varepsilon)\; >\; \backslash delta\; \backslash big)\; =\; -\; \backslash infty.$

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

• {{cite book

| last= Dembo

| first = Amir

|author2=Zeitouni, Ofer

| title = Large deviations techniques and applications

| series = Applications of Mathematics (New York) 38

| edition = Second

| publisher = Springer-Verlag

| location = New York

| year = 1998

| pages = xvi+396

| isbn = 0-387-98406-2

| mr = 1619036

}} (See section 4.2.2)

Category:Asymptotic analysis

Category:Probability theory

Category:Equivalence (mathematics)