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.
Let
(M,d)
M
(\mu\varepsilon)\varepsilon
(\nu\varepsilon)\varepsilon
(\Omega,\Sigma\varepsilon,P\varepsilon)\varepsilon
M
(Y\varepsilon)\varepsilon
(Z\varepsilon)\varepsilon
\varepsilon>0
P\varepsilon
Y\varepsilon
\mu\varepsilon
P\varepsilon
Z\varepsilon
\nu\varepsilon
\delta>0
Y\varepsilon
Z\varepsilon
\delta
\Sigma\varepsilon
\{\omega\in\Omega|d(Y\varepsilon(\omega),Z\varepsilon(\omega))>\delta\}\in\Sigma\varepsilon,
\delta>0
\limsup\varepsilon\varepsilonlogP\varepsilon(d(Y\varepsilon,Z\varepsilon)>\delta)=-infty.
The two families of random variables
(Y\varepsilon)\varepsilon
(Z\varepsilon)\varepsilon
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
I
(\mu\varepsilon)\varepsilon
(\nu\varepsilon)\varepsilon
(\nu\varepsilon)\varepsilon
I