Let be a metric space and consider two one-parameter families of probability measures on , say and . These two families are said to be exponentially equivalent if there exist
a one-parameter family of probability spaces ,
two families of -valued random variables and ,
such that
for each , the -law (i.e. the push-forward measure) of is , and the -law of is ,
for each , " and are further than apart" is a -measurable event, i.e.
for each ,
The two families of random variables and 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 with good rate function, and and are exponentially equivalent, then the same large deviations principle holds for with the same good rate function .
References[]
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. ISBN0-387-98406-2. MR1619036. (See section 4.2.2)