Large Deviations for Markov Chains

Inbunden, Engelska, 2022

Av Alejandro D. de Acosta, Ohio) de Acosta, Alejandro D. (Case Western Reserve University

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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.

Produktinformation

Utgivningsdatum2022-10-27
Mått158 x 235 x 22 mm
Vikt550 g
FormatInbunden
SpråkEngelska
SerieCambridge Tracts in Mathematics
Antal sidor230
FörlagCambridge University Press
ISBN9781316511893

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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 Científicas, University of Wisconsin–Madison, 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.

Preface; 1. Introduction; 2. Lower bounds and a property of lambda; 3. Upper bounds I; 4. Identification and reconciliation of rate functions; 5. Necessary conditions – bounds on the rate function, invariant measures, irreducibility and recurrence; 6. Upper bounds II – equivalent analytic conditions; 7. Upper bounds III – sufficient conditions; 8. The large deviations principle for empirical measures; 9. The case when S is countable and P is matrix irreducible; 10. Examples; 11. Large deviations for vector-valued additive functionals; Appendix A; Appendix B; Appendix C; Appendix D; Appendix E; Appendix F; Appendix G; Appendix H; Appendix I; Appendix J; Appendix K; References; Author index; Subject index.

'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