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Varadhan's lemma

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<sub>ε</sub>) of a family of random variables Z<sub>ε</sub> 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<sub>ε</sub>)<sub>ε&gt;0</sub> be a family of random variables taking values in X; let μ<sub>ε</sub> be the law (probability measure) of Z<sub>ε</sub>. Suppose that (μ<sub>ε</sub>)<sub>ε&gt;0</sub> satisfies the large deviation principle with good rate function I&nbsp;:&nbsp;X&nbsp;→&nbsp;[0,&nbsp;+∞]. Let ϕ &nbsp;:&nbsp;X&nbsp;→&nbsp;R be any continuous function. Suppose that at least one of the following two conditions holds true: either the tail condition

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

Then

See also

References

  • (See theorem 4.3.1)