# General Theory

Chapter

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## Abstract

Throughout this chapter \(\{X^{n}\}_{n\in \mathbb {N}}\) is a sequence of random variables defined on a probability space \((\varOmega ,\mathscr {F}, P)\) and taking values in a complete separable metric space \(\mathscr {X}\). As is usual, we will refer to such a space as a **Polish space** . The metric of \(\mathscr {X}\) is denoted by *d*(*x*, *y*), and expectation with respect to *P* by *E*. The theory of large deviations focuses on random variables \(\{X^{n}\}\) for which the probabilities \(P\{X^{n}\in A\}\) converge to 0 exponentially fast for a class of Borel sets *A*.

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