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 of a family of random variables Zε as ε becomes small in terms of a rate function for the variables.Let X be a regular topological space; let ε>0 be a family of random variables taking values in X; let με be the law of Zε. Suppose that ε>0 satisfies the large deviation principle with good rate function I : X → . Let ϕ : X → R be any continuous function. Suppose that at least one of the following two conditions holds true: either the tail condition
where 1 denotes the indicator function of the event E; or, for some γ > 1, the moment condition
Then
