Varadhan's lemma
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In mathematics, Varadhan's lemma is a result in large deviations theory. The result gives information on the asymptotic distribution of a statistic φ(Zε) of a family of random variables Zε as ε becomes small in terms of a rate function for the variables.
[edit] Statement of the lemma
Let X be a regular topological space; let (Zε)ε>0 be a family of random variables taking values in X; let με be the law (probability measure) of Zε. Suppose that (με)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 \to \infty} \limsup_{\varepsilon \to 0} \varepsilon \log \mathbf{E} \big[ \exp \big( \varphi(Z_{\varepsilon}) / \varepsilon \big) \mathbf{1} \big( \varphi(Z_{\varepsilon}) \geq M \big) \big] = - \infty,](../../../../math/f/8/9/f89e68dfe7a6f9d0031f0cb4ef3134aa.png)
where 1(E) denotes the indicator function of the event E; or, for some γ > 1, the moment condition
![\limsup_{\varepsilon \to 0} \varepsilon \log \mathbf{E} \big[ \exp \big( \gamma \varphi(Z_{\varepsilon}) / \varepsilon \big) \big] < + \infty.](../../../../math/9/d/f/9df5897f2eb3432164c4ef28e493e8d8.png)
Then
![\lim_{\varepsilon \to 0} \varepsilon \log \mathbf{E} \big[ \exp \big( \varphi(Z_{\varepsilon}) / \varepsilon \big) \big] = \sup_{x \in X} \big( \varphi(x) - I(x) \big).](../../../../math/2/3/3/233d2c6a1565225da8b2fadfe1458cf6.png)
[edit] References
- Dembo, Amir; Zeitouni, Ofer (1998). Large deviations techniques and applications, Second edition, Applications of Mathematics (New York) 38, New York: Springer-Verlag, xvi+396. ISBN 0-387-98406-2. MR1619036 (See theorem 4.3.1)
