Varadhan's lemma
In mathematics, Varadhan's lemma is a result from the large deviations theory named after S. R. Srinivasa Varadhan. 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.
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 at least one of the following two conditions holds true: either the tail condition
- [math]\displaystyle{ \lim_{M \to \infty} \limsup_{\varepsilon \to 0} \big(\varepsilon \log \mathbf{E} \big[ \exp\big(\phi(Z_{\varepsilon}) / \varepsilon\big)\,\mathbf{1}\big(\phi(Z_{\varepsilon}) \geq M\big) \big]\big) = -\infty, }[/math]
where 1(E) denotes the indicator function of the event E; or, for some γ > 1, the moment condition
- [math]\displaystyle{ \limsup_{\varepsilon \to 0} \big(\varepsilon \log \mathbf{E} \big[ \exp\big(\gamma \phi(Z_{\varepsilon}) / \varepsilon\big) \big]\big) \lt \infty. }[/math]
Then
- [math]\displaystyle{ \lim_{\varepsilon \to 0} \varepsilon \log \mathbf{E} \big[ \exp\big(\phi(Z_{\varepsilon}) / \varepsilon\big) \big] = \sup_{x \in X} \big( \phi(x) - I(x) \big). }[/math]
See also
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. ISBN 0-387-98406-2. (See theorem 4.3.1)
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