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E-grāmata: Large Deviations for Markov Chains

(Case Western Reserve University, Ohio)
  • Formāts: PDF+DRM
  • Sērija : Cambridge Tracts in Mathematics
  • Izdošanas datums: 27-Oct-2022
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781009063357
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Large Deviations for Markov ChainsPalielināt
  • Formāts: PDF+DRM
  • Sērija : Cambridge Tracts in Mathematics
  • Izdošanas datums: 27-Oct-2022
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781009063357
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This book studies the large deviations for empirical measures and vector-valued additive functionals of Markov chains with general state space. Under suitable recurrence conditions, the ergodic theorem for additive functionals of a Markov chain asserts the almost sure convergence of the averages of a real or vector-valued function of the chain to the mean of the function with respect to the invariant distribution. In the case of empirical measures, the ergodic theorem states the almost sure convergence in a suitable sense to the invariant distribution. The large deviation theorems provide precise asymptotic estimates at logarithmic level of the probabilities of deviating from the preponderant behavior asserted by the ergodic theorems.

The focus of this book is the probability of rare events deviating from the typical long-run behavior of empirical measures and vector-valued additive functionals of general state space Markov chains asserted by the ergodic theorem. The book is of interest to researchers in probability Theory, Statistical Mechanics, and Statistics.

Recenzijas

'This is a treatise on the large deviations for the empirical measure (and additive functionals) of Markov chains. Structural assumptions are kept to a minimum, and great care is taken to separately obtain upper and lower bounds (which under appropriate conditions are shown to be equal). This book summarizes the author's work over the last 40 years and contains a plethora of new, sharpest in its class, results.' Ofer Zeitouni, Weizmann Institute, Israel 'This is a wonderful book. For four decades, Alejandro de Acosta's work in large deviations has been characterized by both its elegance and generality. Large Deviations for Markov Chains synthesizes, refines, and advances the current state of the field in a treatment that is at once rigorous and readable. It belongs on the bookshelf of any mathematician interested in the current state of the subject.' Sandy L. Zabell, Northwestern University 'This is an excellent book, by a top expert in the field, on large deviations for Markov chains in a very general setting. It is a valuable resource for graduate students learning the subject and for researchers in probability theory, statistical mechanics, statistics, engineering, and the sciences.' Firas Rassoul-Agha, University of Utah 'The theory of large deviations is an important way to understand many mathematical and physical models. This book covers the fascinating topic of large deviations for empirical measures and additive functionals of Markov chains with general state space, a subject on which the author is a leading expert who has made crucial contributions. Markov chains represent a large class of stochastic models with a wide spectrum of behaviors. It is remarkable that any universal results, like the ones given in the book, can be formulated for such a large family. It is equally remarkable that the book develops a sharp link between the large deviations and the degree of recurrence of Markov chains. The book does a superb job of clarification, comparison and identification of the rate functions that govern the large deviations.' Xia Chen, University of Tennessee

Papildus informācija

A study of large deviations for empirical measures and vector-valued additive functionals of general state space Markov chains.
Preface xi
1 Introduction
1(13)
1.1 Outline of the Book
11(1)
1.2 Notes
12(2)
2 Lower Bounds and a Property of A
14(22)
2.1 Lower Bounds for Vector-Valued Additive Functionals: The Bounded Case
15(10)
2.2 Lower Bounds for Empirical Measures
25(1)
2.3 The ct(B(S), T(S, iff)) Lower Semicontinuity of A
26(8)
2.4 Notes
34(2)
3 Upper Bounds I
36(11)
3.1 Upper Bounds for Random Probability Measures and Empirical Measures
36(6)
3.2 On the Condition of Exponential Tightness
42(3)
3.3 Notes
45(2)
4 Identification and Reconciliation of Rate Functions
47(21)
4.1 Identification of Rate Functions
48(8)
4.2 On the Relationship between (δ|V)* and Iψ
56(2)
4.3 Reconciliation of Rate Functions
58(6)
4.4 Additional Results on the Equality of Rate Functions
64(2)
4.5 Notes
66(2)
5 Necessary Conditions: Bounds on the Rate Function, Invariant Measures, Irreducibility, and Recurrence
68(11)
5.1 Bounds on the Rate Function J
69(2)
5.2 The V-Tightness of J and Invariant Measures
71(4)
5.3 Irreducibility and Recurrence
75(3)
5.4 Notes
78(1)
6 Upper Bounds II: Equivalent Analytic Conditions
79(15)
6.1 Analytic Properties of Certain Real-Valued Functions on V
79(6)
6.2 Upper Bounds for Random Probability Measures II
85(3)
6.3 Upper Bounds for Empirical Measures II
88(3)
6.4 Necessary Conditions for the Uniformity of a Set of Initial Distributions for the Upper Bound
91(2)
6.5 Notes
93(1)
7 Upper Bounds III: Sufficient Conditions
94(23)
7.1 Sufficient Conditions for the Upper Bound in the V Topology
94(9)
7.2 Some Results When S Is a Polish Space
103(4)
7.3 Another Sufficient Condition for the Upper Bound in the τ Topology
107(8)
7.4 Notes
115(2)
8 The Large Deviation Principle for Empirical Measures
117(17)
8.1 Large Deviations in the V Topology
117(5)
8.2 The Case V = B(S)
122(11)
8.3 Notes
133(1)
9 The Case When S Is Countable and P Is Matrix Irreducible
134(9)
9.1 A Weak Large Deviation Principle
134(1)
9.2 Upper Bounds
135(4)
9.3 The Large Deviation Principle
139(2)
9.4 Notes
141(2)
10 Examples
143(11)
10.1 Different Rate Functions
143(6)
10.2 A Nonconvex Rate Function
149(2)
10.3 A Counterexample
151(2)
10.4 Notes
153(1)
11 Large Deviations for Vector-Valued Additive Functionals
154(43)
11.1 Lower Bounds: The General Case
155(12)
11.2 Identification of the Rate Function δ*∞
167(4)
11.3 Upper Bounds
171(11)
11.4 The Large Deviation Principle
182(3)
11.5 The Zero Set of δ*∞
185(4)
11.6 Sufficient Conditions for the Large Deviation Principle
189(4)
11.7 On the Relationship between Large Deviations for Empirical Measures and Large Deviations for Additive Functionals
193(1)
11.8 Notes
194(3)
Appendix A The Ergodic Theorem for Empirical Measures and Vector-Valued Functionals of a Markov Chain 197(3)
Appendix B Irreducible Kernels, Small Sets, and Petite Sets 200(12)
Appendix C The Convergence Parameter 212(7)
Appendix D Approximation of P by Pt 219(4)
Appendix E On Varadhan's Theorem 223(4)
Appendix F The Duality Theorem for Convex Functions 227(3)
Appendix G Da nidi's Theorem 230(1)
Appendix H Relative Compactness in the V Topology 231(2)
Appendix I A Monotone Class Theorem 233(3)
Appendix J On the Axioms V.1-V.3 and V.1'-V.4 236(4)
Appendix K On Gateaux Differentiability 240(4)
References 244(3)
Author Index 247(1)
Subject Index 248
Alejandro D. de Acosta is Professor Emeritus in the Department of Mathematics, Applied Mathematics and Statistics at Case Western Reserve University. He has taught at the University of California at Berkeley, Massachusetts Institute of Technology, Universidad Nacional de La Plata and Universidad Nacional de Buenos Aires (Argentina), Instituto Venezolano de Investigaciones Cientķficas, University of WisconsinMadison, and, since 1983, at Case Western Reserve University. He is a Fellow of the Institute of Mathematical Statistics, and has served on the editorial boards of the Annals of Probability and the Journal of Theoretical Probability. He has published research papers in a number of areas of Probability Theory.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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