Large Deviations for Additive Functionals of Markov Chains
Author: Alejandro D. de Acosta
Publisher: American Mathematical Soc.
Published: 2014-03-05
Total Pages: 120
ISBN-13: 0821890891
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Large Deviations for Markov Chains
Author: Alejandro D. de Acosta
Publisher:
Published: 2022-10-12
Total Pages: 264
ISBN-13: 1009063359
DOWNLOAD EBOOK →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.
Large Deviations for Additive Functionals of Markov Exhangeable Sequences
Local Limit Theorems for Inhomogeneous Markov Chains
Author: Dmitry Dolgopyat
Publisher: Springer Nature
Published: 2023-07-31
Total Pages: 348
ISBN-13: 3031326016
DOWNLOAD EBOOK →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.
On the Large Deviation Functions of Markov Chains
Author: University of Minnesota. Institute for Mathematics and Its Applications
Publisher:
Published: 1985
Total Pages: 20
ISBN-13:
DOWNLOAD EBOOK →Asymptotics of Large Deviations for I.I.D. and Markov Additive Random Variables in R[superscript D]
Large Deviations and Applications
Author: S. R. S. Varadhan
Publisher: SIAM
Published: 1984-01-31
Total Pages: 74
ISBN-13: 0898711894
DOWNLOAD EBOOK →Many situations exist in which solutions to problems are represented as function space integrals. Such representations can be used to study the qualitative properties of the solutions and to evaluate them numerically using Monte Carlo methods. The emphasis in this book is on the behavior of solutions in special situations when certain parameters get large or small.
Large Deviations
Author: S. R. S. Varadhan
Publisher: American Mathematical Soc.
Published: 2016-12-08
Total Pages: 114
ISBN-13: 082184086X
DOWNLOAD EBOOK →The theory of large deviations deals with rates at which probabilities of certain events decay as a natural parameter in the problem varies. This book, which is based on a graduate course on large deviations at the Courant Institute, focuses on three concrete sets of examples: (i) diffusions with small noise and the exit problem, (ii) large time behavior of Markov processes and their connection to the Feynman-Kac formula and the related large deviation behavior of the number of distinct sites visited by a random walk, and (iii) interacting particle systems, their scaling limits, and large deviations from their expected limits. For the most part the examples are worked out in detail, and in the process the subject of large deviations is developed. The book will give the reader a flavor of how large deviation theory can help in problems that are not posed directly in terms of large deviations. The reader is assumed to have some familiarity with probability, Markov processes, and interacting particle systems.
Large Deviations of Uniformly Recurrent Markov Additive Processes
An Introduction to the Theory of Large Deviations
Author: D.W. Stroock
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 204
ISBN-13: 1461385148
DOWNLOAD EBOOK →These notes are based on a course which I gave during the academic year 1983-84 at the University of Colorado. My intention was to provide both my audience as well as myself with an introduction to the theory of 1arie deviations • The organization of sections 1) through 3) owes something to chance and a great deal to the excellent set of notes written by R. Azencott for the course which he gave in 1978 at Saint-Flour (cf. Springer Lecture Notes in Mathematics 774). To be more precise: it is chance that I was around N. Y. U. at the time'when M. Schilder wrote his thesis. and so it may be considered chance that I chose to use his result as a jumping off point; with only minor variations. everything else in these sections is taken from Azencott. In particular. section 3) is little more than a rewrite of his exoposition of the Cramer theory via the ideas of Bahadur and Zabel. Furthermore. the brief treatment which I have given to the Ventsel-Freidlin theory in section 4) is again based on Azencott's ideas. All in all. the biggest difference between his and my exposition of these topics is the language in which we have written. However. another major difference must be mentioned: his bibliography is extensive and constitutes a fine introduction to the available literature. mine shares neither of these attributes. Starting with section 5).
