User:Gc12/sandbox
Appearance
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.
Statement of the theorem for real variables
[edit]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
We assume that there exists a sequence an converging to 0, such that the limit function
exists, as an element of the extended real line Λ(s) ∈[-∞ ∞].
See also
[edit]- 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 Rd
- Schilder's theorem, a large deviations principle for Brownian motion
- Varadhan's lemma
- Extreme value theory
- Large deviations of Gaussian random functions
References
[edit]- ^ J. Gärtner, On large deviations from the invariant measure, Th. Probab. Appl. 22 (1977),24-39.
- ^ R.S. Ellis, Large deviations for a general class of random vectors, An. Probab. 12 (1984),1-12.
Bibliography
[edit]- Large Deviations Techniques and Applications by Amir Dembo and Ofer Zeitouni. Springer ISBN 0-387-98406-2
External links
[edit]