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_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

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

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

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

Then

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

See also

• Laplace principle (large deviations theory)

References

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