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(Zε) of a family of random variables Zε 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 (Zε)ε>0 be a family of random variables taking values in X; let με be the law (probability measure) of Zε. Suppose that (με)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 → ∞ lim sup ε → 0 ( ε log ⁡ E [ exp ⁡ ( ϕ ( Z ε ) / ε ) 1 ( ϕ ( Z ε ) ≥ M ) ] ) = − ∞ , {\displaystyle \lim _{M\to \infty }\limsup _{\varepsilon \to 0}{\big (}\varepsilon \log \mathbf {E} {\big [}\exp {\big (}\phi (Z_{\varepsilon })/\varepsilon {\big )}\,\mathbf {1} {\big (}\phi (Z_{\varepsilon })\geq M{\big )}{\big ]}{\big )}=-\infty ,}

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

lim sup ε → 0 ( ε log ⁡ E [ exp ⁡ ( γ ϕ ( Z ε ) / ε ) ] ) < ∞ . {\displaystyle \limsup _{\varepsilon \to 0}{\big (}\varepsilon \log \mathbf {E} {\big [}\exp {\big (}\gamma \phi (Z_{\varepsilon })/\varepsilon {\big )}{\big ]}{\big )}<\infty .}

Then

lim ε → 0 ε log ⁡ E [ exp ⁡ ( ϕ ( Z ε ) / ε ) ] = sup x ∈ X ( ϕ ( x ) − I ( x ) ) . {\displaystyle \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

  • 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. MR 1619036. (See theorem 4.3.1)