Varadhan's lemma

In mathematics, Varadhan's lemma is a result in large deviations theory. The result gives information on the asymptotic distribution of a statistic "φ"("Z"_"ε") of a family of random variables "Z"_"ε" as "ε" becomes small in terms of a rate function for the variables.

tatement of the lemma

Let "X" be a regular topological space; let ("Z"_"ε")_"ε">0 be a family of random variables taking values in "X"; let "μ"_"ε" be the law (probability measure) of "Z"_"ε". Suppose that ("μ"_"ε")_"ε">0 satisfies the large deviation principle with good rate function "I" : "X" → [0, +∞] . Let "φ" : "X" → R be any continuous function. Suppose that either one of the following two conditions holds true: either the tail condition

: $lim_{M o infty} limsup_{varepsilon o 0} varepsilon log mathbf{E} ig [ exp ig( varphi(Z_{varepsilon}) / varepsilon ig) mathbf{1} ig( varphi(Z_{varepsilon}) geq M ig) ig] = - infty,$

where 1("E") denotes the indicator function of the event "E"; or, for some "γ" > 1, the moment condition

: $limsup_{varepsilon o 0} varepsilon log mathbf{E} ig [ exp ig( gamma varphi(Z_{varepsilon}) / varepsilon ig) ig] < + infty.$

Then

: $lim_{varepsilon o 0} varepsilon log mathbf{E} ig [ exp ig( varphi(Z_{varepsilon}) / varepsilon ig) ig] = sup_{x in X} ig( varphi(x) - I(x) ig).$

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 theorem 4.3.1)

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