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)

Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

Large deviations theory — In Probability Theory, the Large Deviations Theory concerns the asymptotic behaviour of remote tails of sequences of probability distributions. Some basic ideas of the theory can be tracked back to Laplace and Cramér, although a clear unified… … Wikipedia
List of mathematics articles (V) — NOTOC Vac Vacuous truth Vague topology Valence of average numbers Valentin Vornicu Validity (statistics) Valuation (algebra) Valuation (logic) Valuation (mathematics) Valuation (measure theory) Valuation of options Valuation ring Valuative… … Wikipedia
Laplace principle (large deviations theory) — In mathematics, Laplace s principle is a basic theorem in large deviations theory, similar to Varadhan s lemma. It gives an asymptotic expression for the Lebesgue integral of exp(− theta; phi; ( x )) over a fixed set A as theta; becomes large.… … Wikipedia
Tilted large deviation principle — In mathematics mdash; specifically, in large deviations theory mdash; the tilted large deviation principle is a result that allows one to generate a new large deviation principle from an old one by tilting , i.e. integration against an… … Wikipedia
List of lemmas — This following is a list of lemmas (or, lemmata , i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures. 0 to 9 *0/1 Sorting Lemma ( comparison… … Wikipedia
Ito Kiyoshi — Itō Kiyoshi an der Cornell University, 1970 Itō Kiyoshi (jap. 伊藤清; * 7. September 1915 in Hokusei chō (heute Inabe), Präfektur Mie; † 10. November 2008 in Kyōto) war ein japanischer … Deutsch Wikipedia
Kiyoshi Ito — Itō Kiyoshi an der Cornell University, 1970 Itō Kiyoshi (jap. 伊藤清; * 7. September 1915 in Hokusei chō (heute Inabe), Präfektur Mie; † 10. November 2008 in Kyōto) war ein japanischer … Deutsch Wikipedia
Kiyoshi Itō — Itō Kiyoshi an der Cornell University, 1970 Itō Kiyoshi (jap. 伊藤清; * 7. September 1915 in Hokusei chō (heute Inabe), Präfektur Mie; † 10. November 2008 in Kyōto) war ein japanischer … Deutsch Wikipedia
Kiyosi Itô — Itō Kiyoshi an der Cornell University, 1970 Itō Kiyoshi (jap. 伊藤清; * 7. September 1915 in Hokusei chō (heute Inabe), Präfektur Mie; † 10. November 2008 in Kyōto) war ein japanischer … Deutsch Wikipedia
Itō Kiyoshi — an der Cornell University, 1970 Itō Kiyoshi (jap. 伊藤清; * 7. September 1915 in Hokusei chō (heute Inabe), Präfektur Mie; † 10. November 2008 in Kyōto) war ein japanischer … Deutsch Wikipedia

Academic Dictionaries and Encyclopedias

Varadhan's lemma

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Varadhan's lemma

Look at other dictionaries:

Share the article and excerpts

Direct link