MATH 585 - TOPICS IN MATHEMATICAL PHYSICS - FALL 2006 MATHEMATICS OF MEAN FIELD SPIN GLASSES AND THE REPLICA METHOD LECTURE 5: SANOV'S AND CRAM ´ ER'S THEOREMS
Shannon L. Starr, Stone-Weierstrass Theorem · 2006
In this lecture we will review the most elementary aspects of large deviations theory. This is the large deviation theory for i.i.d. random variables taking only finitely many values. Following Dembo and Zeitouni [2], we start with Sanov’s theorem. We will basically reproduce Dembo and Zeitouni’s proof of Sanov’s theorem, and its corollary, Cramer’s theorem. The interested reader is referred to their book for many deeper results. Let Ω = {a1, . . . , an} be a finite set. This is simply a finite sample space. But Dembo and Zeitouni also refer to it as a finite alphabet, presumably because of the importance of Sanov’s theorem in information theory. Let M1(Ω) be the space of probability measures on Ω. Every such measure can be written as