By Richard S. Ellis
From the reports: "... every one bankruptcy of the e-book is by way of a notes part and by means of a difficulties part. There are over a hundred difficulties, lots of that have tricks. The e-book could be suggested as a textual content, it offers a completly self-contained studying ..." --S. Pogosian in Zentralblatt für Mathematik
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Extra resources for Entropy, Large Deviations, and Statistical Mechanics (Classics in Mathematics)
Sample text
X,(ojlX,iojlX,(col.. ) in Q; (X(n,co))^ = X^(co), (X(n,cjo))2 = ^li^)^ probability measure on ^(Q) by (1-33) ^tc. For each COGQ, define a ^>,-) = ^"f Vm«)(-), where T^ is the identity mapping and T^ = T(T^~^) for /c = 2 , . . , « - 1. 6. Level-3: Empirical Process 23 each Borel subset B of Q, R„(a),B) is the relative frequency with which X(n,(jo), TX{n,co), . . , r""^X(w, co) is in B. oS) is periodic of period n, R(n, •) is for each co a strictly stationary probabiHty measure. It is called the empirical process corresponding to A^i(co), .
These generalize essentially without change. But they also have additional features that are worth emphasizing. The first theorem states the contraction principle relating levels-1 and 2. * Let p be a Borelprobability measure on U^ such that Cp{t) = is finite for all teU^. 18). 21) Il'\z) = inf |/^^>(v): v G J^(U'l | xv(dx) = z\ . 1 is proved in Donsker-Varadhan (1976a, page 425). 1] under additional hypotheses on p. 5. Contraction Principles 43 fimum on the right-hand side is attained at v = p (Il^\p) = 0).
The last section of this chapter shows how one can apply large deviations to evaluate the asymptotics of certain configuration space integrals that depend on a parameter. The main theorem in this section is due to Varadhan (1966). In the next few chapters we will apply Varadhan's theorem to study integrals associated with the statistical mechanical models (partition functions). 2. D. Random Vectors The presentation of the three levels of large deviations for vector-valued random variables involves some topology.