# Introduction to Large Deviation Theory

# Introduction to Large Deviation Theory

This chapter provides an introduction to large deviation theory. It begins with an overview of the motivatio n for the problem under study, focusing on probability distributions and how to construct an empirical distribution. It then considers the notion of a lower semi-continuous function and that of a lower semi-continuous relaxation before discussing the large deviation property for i.i.d. samples. In particular, it describes Sanov's theorem for a finite alphabet and proceeds by analyzing large deviation property for Markov chains, taking into account stationary distributions, entropy and relative entropy rates, the rate function for doubleton frequencies, and the rate function for singleton frequencies.

*Keywords:*
large deviation theory, probability distribution, lower semi-continuous function, lower semi-continuous relaxation, large deviation property, Sanov's theorem, Markov chain, stationary distribution, relative entropy rate, rate function

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