Freidlin–Wentzell theorem

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

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