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)