Local Limit Theorems for Inhomogeneous Markov Chains

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.

- 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. 

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.