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In probability theory, the Gärtner-Ellis theorem is a result concerning large deviations for sequences of random variables, due to Gärtner ^[1], and Ellis ^[2] The theorem generalises several well known results, including Cramér's theorem, and Sanov's theorem.

Let X_n be a sequence of random vectors taking values in R^d, and suppose that X_n has law μ_n. We define the cumulant generating function, or log-moment generating function, of X_n to be

${\displaystyle \Lambda _{N}(s)=\log \mathbb {E} \left[e^{\langle s,X_{n}\rangle }\right],\qquad s\in \mathbb {R} ^{d}.}$

We assume that there exists a sequence a_n converging to 0, such that the limit function

${\displaystyle \Lambda (s)=\lim _{n\rightarrow \infty }a_{n}\Lambda _{n}(a_{n}^{-1}s),}$

exists, as an element of the extended real line Λ(s) ∈[-∞ ∞].

• Cramér's large deviation theorem
• Chernoff's inequality
• Contraction principle (large deviations theory), a result on how large deviations principles "push forward"
• Freidlin–Wentzell theorem, a large deviations principle for Itō diffusions
• Laplace principle, a large deviations principle in R^d
• Schilder's theorem, a large deviations principle for Brownian motion
• Varadhan's lemma
• Extreme value theory
• Large deviations of Gaussian random functions

• Large Deviations Techniques and Applications by Amir Dembo and Ofer Zeitouni. Springer ISBN 0-387-98406-2

• An elementary introduction to the Large Deviations Theory