Varadhan's lemma
In mathematics, Varadhan's lemma is a result from 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
![{\displaystyle \lim _{M\to \infty }\limsup _{\varepsilon \to 0}\varepsilon \log \mathbf {E} {\big [}\exp {\big (}\phi (Z_{\varepsilon })/\varepsilon {\big )}\mathbf {1} {\big (}\phi (Z_{\varepsilon })\geq M{\big )}{\big ]}=-\infty ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4327a6c9d1eefc2832e9466fb2670a90163d6de)
where 1(E) denotes the indicator function of the event E; or, for some γ > 1, the moment condition
![{\displaystyle \limsup _{\varepsilon \to 0}\varepsilon \log \mathbf {E} {\big [}\exp {\big (}\gamma \phi (Z_{\varepsilon })/\varepsilon {\big )}{\big ]}<+\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f835e9802cd6f104a1b55e9a12a161a886336e8e)
Then
![{\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 )}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eba100808619b3d7847eb4c143dacb949c6ac093)
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. MR 1619036. (See theorem 4.3.1)