# Large Deviations in Ergodic Theory

• Steven Orey
Chapter
Part of the Progress in Probability and Statistics book series (PRPR, volume 9)

## Abstract

The classical example of a large deviation result is Cramer’s theorem. It tells us, in a contemporary formulation, that if Y1, Y2,… is a sequence of independent real valued random variables with identical distribution function F such that
$$f(\theta ) = E[\exp \{ \theta Y_1 \} [ = \smallint \exp \{ \theta y\} F(dy)$$
is finite for all finite θ,and if Zn = (Y1) + Y2 + … Yn/n then
$$\text{k}(\text{x})\, = \,\mathop {\sup }\limits_\theta \,[\theta x\, - \,\log \,f(\theta )]$$
satisfies
1. (0.1)

$$\overline {_{n \to \infty }^{\lim } } \frac{1} {\text{n}}\,\log \,\text{P}[z_n \, \in \text{A}]\, \leqslant \text{ } - \inf \text{ }k(a):a \in \text{A}$$ A closed

and
1. (0.2)

$$\overline {_{n \to \infty }^{\lim } } \frac{1}{\text{n}}\,\log \,\text{P}[z_n \, \in \text{A}]\, \geqslant \text{ } - \inf \{ k(a):a \in \text{A}\}$$ A Open.

## Keywords

Ergodic Theory Compactness Condition Deviation Function Ergodic Theorem Homomorphic Image
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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