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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, and Ellis The theorem generalises several well known results, including Cramér's theorem, and Sanov's theorem.

Statement of the theorem for real variables
Let Xn be a sequence of random vectors taking values in Rd, and suppose that Xn has law μn. We define the cumulant generating function, or log-moment generating function, of Xn to be
 * $$ \Lambda_N(s) = \log \mathbb{E} \left[ e^{\langle s, X_n \rangle} \right], \qquad s \in \mathbb R^d. $$

We assume that there exists a sequence an converging to 0, such that the limit function
 * $$ \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) &isin;[-∞ ∞].