Applications-The Finite Dimensional Case.

This chapter consists of applications of the theory presented in Chapter 2. The LDPs associated with finite state irreducible Markov chains are derived in Section 3.1 as a corollary of the Gärtner–Ellis theorem. Varadhan΄s characterization of the spectral radius of nonnegative irreducible matrices is derived along the way. (See Exercise 3.1.19.) The asymptotic size of long rare segments in random walks is found by combining, in Section 3.2, the basic large deviations estimates of Cramér΄s theorem with the Borel–Cantelli lemma. The Gibbs conditioning principle is of fundamental importance in statistical mechanics. It is derived in Section 3.3, for finite alphabet, as a direct result of Sanov΄s theorem. The asymptotics of the probability of error in hypothesis testing problems are analyzed in Sections 3.4 and 3.5 for testing between two a priori known product measures and for universal testing, respectively. Shannon΄s source coding theorem is proved in Section 3.6 by combining the classical random coding argument with the large deviations lower bound of the Gärtner–Ellis theorem. Finally, Section 3.7 is devoted to refinements of Cramér΄s theorem in ℝ. Specifically, it is shown that for βϵ(0,1/2), satisfies the LDP with a Normal-like rate function, and the pre-exponent associated with is computed for appropriate values of q. [ABSTRACT FROM AUTHOR]