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In this chapter, we revisit three applications considered in Chapters 2 and 3 in the finite alphabet setup. Equipped with Sanov΄s theorem and the projective limit approach, the general case (Σ Polish) is treated here. [ABSTRACT FROM AUTHOR]

One of the striking successes of the large deviations theory in the setting of finite dimensional spaces explored in Chapter 2 was the ability to obtain refinements, in the form of Cramér΄s theorem and the Gärtner–Ellis theorem, of the weak law of large numbers. As demonstrated in Chapter 3, this particular example of an LDP leads to many important applications; and in this chapter, the problem is tackled again in a more general setting, moving away from the finite dimensional world. [ABSTRACT FROM AUTHOR]

The finite dimensional LDPs considered in Chapter 2 allow computations of the tail behavior of rare events associated with various sorts of empirical means. In many problems, the interest is actually in rare events that depend on a collection of random variables, or, more generally, on a random process. Whereas some of these questions may be cast in terms of empirical measures, this is not always the most fruitful approach. Interest often lies in the probability that a path of a random process hits a particular set. Questions of this nature are addressed in this chapter. In Section 5.1, the case of a random walk, the simplest example of all, is analyzed. The Brownian motion counterpart is then an easy application of exponential equivalence, and the diffusion case follows by suitable approximate contractions. The range of applications presented in this chapter is also representative: stochastic dynamical systems (Sections 5.4, 5.7, and 5.8), DNA matching problems and statistical change point questions (Section 5.5). [ABSTRACT FROM AUTHOR]

In this chapter, we initiate the investigation of LDPs for families of measures on general spaces. As will be obvious in subsequent chapters, the objects on which the LDP is sought may vary considerably. Hence, it is necessary to undertake a study of the LDP in an abstract setting. We shall focus our attention on the abstract statement of the LDP as presented in Section 1.2 and give conditions for the existence of such a principle and various approaches for the identification of the resulting rate function. [ABSTRACT FROM AUTHOR]

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]

This chapter is devoted to the study of the LDP in a framework that is not yet encumbered with technical details. The main example studied is the empirical mean of a sequence of random variables taking values in ℝd. The concreteness of this situation enables the LDP to be obtained under conditions that are much weaker than those that will be imposed in the ˵general″ theory. Many of the results presented here have counterparts in the infinite dimensional context dealt with later, starting in Chapter 4. [ABSTRACT FROM AUTHOR]

This book is concerned with the study of the probabilities of very rare events. In this introduction, the notion of rare is quantified, and the Large Deviation Principle (LDP) is introduced. [ABSTRACT FROM AUTHOR]