Exponentially equivalent measures explained

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

and

(\nu\varepsilon)\varepsilon

. These two families are said to be exponentially equivalent if there exist

(\Omega,\Sigma\varepsilon,P\varepsilon)\varepsilon

,

M

-valued random variables

(Y\varepsilon)\varepsilon

and

(Z\varepsilon)\varepsilon

,such that

\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

,

\delta>0

, "

Y\varepsilon

and

Z\varepsilon

are further than

\delta

apart" is a

\Sigma\varepsilon

-measurable event, i.e.

\{\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

and

(Z\varepsilon)\varepsilon

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

with good rate function

I

, and

(\mu\varepsilon)\varepsilon

and

(\nu\varepsilon)\varepsilon

are exponentially equivalent, then the same large deviations principle holds for

(\nu\varepsilon)\varepsilon

with the same good rate function

I

.

References