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(Z_ε) of a family of random variables Z_ε 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 (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 at least one of the following two conditions holds true: either the tail condition

$\lim _{{M\to \infty }}\limsup _{{\varepsilon \to 0}}\varepsilon \log {\mathbf {E}}{\big [}\exp {\big (}\phi (Z_{{\varepsilon }})/\varepsilon {\big )}{\mathbf {1}}{\big (}\phi (Z_{{\varepsilon }})\geq M{\big )}{\big ]}=-\infty ,$

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

$\limsup _{{\varepsilon \to 0}}\varepsilon \log {\mathbf {E}}{\big [}\exp {\big (}\gamma \phi (Z_{{\varepsilon }})/\varepsilon {\big )}{\big ]}<+\infty .$

Then

$\lim _{{\varepsilon \to 0}}\varepsilon \log {\mathbf {E}}{\big [}\exp {\big (}\phi (Z_{{\varepsilon }})/\varepsilon {\big )}{\big ]}=\sup _{{x\in X}}{\big (}\phi (x)-I(x){\big )}.$

See also

• Laplace principle (large deviations theory)

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)