A Course on Large Deviations with an Introduction to Gibbs Measures.

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$Lecture notes for Math 522 Spring 2012 (Rudin chapter 7)$

$Math 320 - Fall 2010 HW #7 Selected Solutions Problem 5.1.32 Let ...$

32 4. Yet some more generalities Varadhan’s theorem. Suppose LDP(µn, rn, I) holds, f : X → [−∞, ∞) is a continuous function, and lim b→∞ lim 1 n→∞ rn log e f≥b rnf (4.1) dµn = −∞. Then 1 lim n→∞ rn log e rnf dµn = sup(f − I). Exercise 4.6. Show that condition (4.1) is satisfied when there exists an α > 1 such that sup n e αrnf 1/rn dµn < ∞. This in turn is satisfied, for example, when f is bounded above. The general idea behind the proof of the theorem is similar to the one used when µn = µ is independent of n. Namely, if one partitions the space into sets Ui on which µn(Ui) ∼ e−rnI(xi) for xi ∈ Ui, then 1 rn log e rnf dµn ∼ 1 rn log e nf(xi) µn(Ui) ∼ 1 rn log e n[f(xi)−I(xi)] ∼ max[f(xi) − I(xi)]. The actual proof follows from the next two lemmas. Lemma 4.7. Let f : X → [−∞, ∞] be a lower semicontinuous function. Assume {µn} satisfies the lower large deviation bound (2.4) with rate function I and normalization rn. Then, 1 lim rn n→∞ log e rnf dµn ≥ sup (f − I). f∧I −∞. Let −∞ < c < f(x). Then G = {f > c} contains x, is open, and is such that infG f ≥ c. Write 1 lim rn n→∞ log e rnf dµn ≥ lim n→∞ 1 rn log ≥ c + lim n→∞ 1 rn G i e rnf dµn log µn(G) ≥ c − inf G I ≥ c − I(x). Now let c grow to f(x) then take sup over x. Lemma 4.8. Let f : X → [−∞, ∞] be an upper semicontinuous function (i.e. −f is lower semicontinuous). Assume {µn} satisfies the upper large
4.2. Varadhan’s theorem and Bryc’s theorem 33 deviation bound (2.3) with rate function I and normalization rn. Assume also that lim b→∞ lim 1 n→∞ rn log e f≥b rnf dµn = −∞. Then, 1 lim n→∞ rn log e rnf dµn ≤ sup (f − I). f∧I
Page 1 and 2: A Course on Large
Page 3: To Alla, Maxim, and Kirill To Celes
Page 6 and 7: viii Contents §5.3. Multidimension
Page 9 and 10: Preface (and to the reviewers) This
Page 11: Preface (and to the reviewers) xiii
Page 15 and 16: Introduction Chapter 1 Imagine the
Page 17 and 18: 1.1. Information-theoretic entropy
Page 19 and 20: 1.2. Thermodynamic entropy 7 to 0.
Page 21: 1.2. Thermodynamic entropy 9 The id
Page 24 and 25: 12 2. Preliminary examples and gene
Page 32 and 33: 20 3. More generalities and Cramér
Page 42 and 43: 30 4. Yet some more generalities Th
Page 46 and 47: 34 4. Yet some more generalities Va
Page 48 and 49: 36 4. Yet some more generalities fu
Page 50 and 51: 38 4. Yet some more generalities Ha
Page 52 and 53: 40 4. Yet some more generalities Ne
Page 54 and 55: 42 5. Convex analysis in large devi
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Page 84 and 85: 72 7. Large deviations for i.i.d. f
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82 7. Large deviations for i.i.d. f
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84 7. Large deviations for i.i.d. f
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Formalism for classical lattice sys
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8.2. Potentials and Hamiltonians 89
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8.3. Specifications 91 Example 8.5.
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8.4. Gibbs specifications and phase
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8.4. Gibbs specifications and phase
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8.5. Observables 97 In either case,
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Large deviations and equilibrium st
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9.2. Thermodynamic limit of the pre
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9.3. Specific relative entropy 103
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9.5. Dobrushin-Lanford-Ruelle (DLR)
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112 10. Phase transition in the Isi
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Percolation approach to phase trans
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Further asymptotics for i.i.d. rand
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12.1. Refinement of Cramér’s the
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12.2. Moderate deviations 137 Theor
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12.2. Moderate deviations 139 where
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142 13. Large deviations for Markov
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144 13. Large deviations for Markov
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146 14. Convexity criterion for lar
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Nonstationary independent variables
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15.2. Proof of the large deviation
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Topics from probability Appendix A
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A.1. Weak convergence of probabilit
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A.2. Ergodic theorem 171 To illumin
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A.2. Ergodic theorem 173 measurable
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A.3. Stochastic ordering 175 de Fin
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Topics from analysis Ultimately, th
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B.2. Minimax theorem 179 ∗ Exerci
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B.2. Minimax theorem 181 Now, if
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184 C. Inequalities Next, let C = m
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186 C. Inequalities The inequality
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Bibliography 1. Michael Aizenman, I
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Bibliography 191 24. Y. Higuchi, On
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Notation index empirical mean (X1 +
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Notation index 195 B space of absol
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Aizenman, 113 Bartle, 6 Baxter and
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202 General index average, 99 field
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204 General index proper function,
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A Course on Large Deviations with an Introduction to Gibbs Measures.

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