Varadhan's lemma explained

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(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} \limsup _{\varepsilon} ( \varepsilon log E [ \exp ( \phi(Z _{\varepsilon} ) / \varepsilon ) 1 ( \phi(Z _{\varepsilon} ) \geq M ) ] ) = -inf ty,

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

\limsup _{\varepsilon} ( \varepsilon log E [ \exp ( \gamma \phi(Z _{\varepsilon} ) / \varepsilon ) ] ) < inf ty.

Then

\lim _{\varepsilon} \varepsilon log E [ \exp ( \phi(Z _{\varepsilon} ) / \varepsilon ) ] = \sup _{x} ( \phi(x) - I(x) ) .

See also

• Laplace principle (large deviations theory)

References

• Book: Dembo , Amir . Zeitouni, Ofer . Large deviations techniques and applications. Applications of Mathematics (New York) 38. Second. Springer-Verlag. New York. 1998. xvi+396. 0-387-98406-2. 1619036. (See theorem 4.3.1)