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

$\underset{M\to \mathrm{\infty }}{\lim }{\mathrm{lim\; sup}}_{\epsilon \to 0}\epsilon \log \mathbf{E}[\exp (\varphi (Z_{\epsilon })/\epsilon \left)\mathbf{1}\right(\varphi (Z_{\epsilon })\ge M\left)\right]=-\mathrm{\infty },$

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

${\mathrm{lim\; sup}}_{\epsilon \to 0}\epsilon \log \mathbf{E}[\exp (\gamma \varphi (Z_{\epsilon })/\epsilon \left)\right]<+\mathrm{\infty }.$

Then

$\underset{\epsilon \to 0}{\lim }\epsilon \log \mathbf{E}[\exp (\varphi (Z_{\epsilon })/\epsilon \left)\right]={\sup }_{x\in X}(\varphi (x)-I(x)).$

See also

• Laplace principle (large deviations theory)

References

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