Exponentially equivalent measures
- Exponentially equivalent measures
In mathematics, the notion of exponential equivalence of measures is a concept that describes 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 ("μ""ε")"ε">0 and ("ν""ε")"ε">0. These two families are said to be exponentially equivalent if there exist
* a one-parameter family of probability spaces ((Ω, Σ"ε", P"ε"))"ε>0,
* two families of "M"-valued random variables ("Y""ε")"ε">0 and ("Z""ε")"ε">0,such that
* for each "ε" > 0, the P"ε"-law (i.e. the push-forward measure) of "Y""ε" is "μ""ε", and the P"ε"-law of "Z""ε" is "ν""ε",
* for each "δ" > 0, “"Y""ε" and "Z""ε" are further than "δ" apart” is a Σ"ε"-measurable event, i.e.::
* for each "δ" > 0,::
The two families of random variables ("Y""ε")"ε">0 and ("Z""ε")"ε">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 ("μ""ε")"ε">0 with good rate function "I", and ("μ""ε")"ε">0 and ("ν""ε")"ε">0 are exponentially equivalent, then the same large deviations principle holds for ("ν""ε")"ε">0 with the same good rate function "I".
References
* cite book
last= Dembo
first = Amir
coauthors = Zeitouni, Ofer
title = Large deviations techniques and applications
series = Applications of Mathematics (New York) 38
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 1998
pages = xvi+396
isbn = 0-387-98406-2 MathSciNet|id=1619036 (See section 4.2.2)
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