Local Limit Theorems for Inhomogeneous Markov Chains

Dmitry Dolgopyat, Omri M. Sarig (Autoren)

Buch | Softcover

XIII, 342 Seiten

2023 | 1st ed. 2023
Springer International Publishing (Verlag)
978-3-031-32600-4 (ISBN)

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This book extends the local central limit theorem to Markov chains whose state spaces and transition probabilities are allowed to change in time. Such chains are used to model Markovian systems depending on external time-dependent parameters. The book develops a new general theory of local limit theorems for additive functionals of Markov chains, in the regimes of local, moderate, and large deviations, and provides nearly optimal conditions for the classical expansions, as well as asymptotic corrections when these conditions fail. Applications include local limit theorems for independent but not identically distributed random variables, Markov chains in random environments, and time-dependent perturbations of homogeneous Markov chains.
The inclusion of appendices with background material, numerous examples, and an account of the historical background of the subject make this self-contained book accessible to graduate students. It will also be useful for researchers in probability and ergodic theory who are interested in asymptotic behaviors, Markov chains in random environments, random dynamical systems and non-stationary systems.

lt;b> Dmitry Dolgopyat is a Distiguished Professor at the University of Maryland, and a member of the advisory board of the Brin Mathematics Research Center. He obtained his doctorate from Princeton University, and has held positions at the University of California at Berkeley, and at the Pennsylvania State University.

Omri Sarig is the Theodore R. and Edlyn Racoosin Professor of Mathematics at the Weizmann Institute of Science. He obtained his doctorate from Tel-Aviv University, and has held positions at the University of Warwick and the Pennsylvania State University.

Dmitry Dolgopyat and Omri Sarig work at the intersection of ergodic theory, probability theory, and the theory of dynamical systems. They have an ongoing collaboration aimed at studying probabilistic limit theorems for dynamical systems.

- 1. Overview. - 2. Markov Arrays, Additive Functionals, and Uniform Ellipticity. - 3. Variance Growth, Center-Tightness, and the Central Limit Theorem. - 4. The Essential Range and Irreducibility. - 5. The Local Limit Theorem in the Irreducible Case. - 6. The Local Limit Theorem in the Reducible Case. - 7. Local Limit Theorems for Moderate Deviations and Large Deviations. - 8. Important Examples and Special Cases. - 9. Local Limit Theorems for Markov Chains in Random Environments.

Erscheinungsdatum	29.06.2023
Reihe/Serie	Lecture Notes in Mathematics
Zusatzinfo	XIII, 342 p. 1 illus. in color.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	545 g
Themenwelt	Mathematik / Informatik ► Informatik ► Theorie / Studium
	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
	Technik ► Maschinenbau
Schlagworte	Asymptotic behaviour • central limit theorem • Dynamical Systems • ergodic theory • Inhomogeneous • large deviations • Local Limit Theorem • markov chains • random dynamical systems • Twisted Transfer Operators
ISBN-10	3-031-32600-8 / 3031326008
ISBN-13	978-3-031-32600-4 / 9783031326004
Zustand	Neuware