Approximating Countable Markov Chains

ISBN-13: 9781461382324 / Angielski / Miękka / 2011 / 140 str.

David Freedman

Opis

A long time ago I started writing a book about Markov chains, Brownian motion, and diffusion. I soon had two hundred pages of manuscript and my publisher was enthusiastic. Some years and several drafts later, I had a thousand pages of manuscript, and my publisher was less enthusiastic. So we made it a trilogy: Markov Chains Brownian Motion and Diffusion Approximating Countable Markov Chains familiarly - MC, B & D, and ACM. I wrote the first two books for beginning graduate students with some knowledge of probability; if you can follow Sections 10.4 to 10.9 of Markov Chains, you're in. The first two books are quite independent of one another, and completely independent of this one, which is a monograph explaining one way to think about chains with instantaneous states. The results here are supposed to be new, except when there are specific disclaimers. It's written in the framework of Markov chains; we wanted to reprint in this volume the MC chapters needed for reference. but this proved impossible. Most of the proofs in the trilogy are new, and I tried hard to make them explicit. The old ones were often elegant, but I seldom saw what made them go. With my own, I can sometimes show you why things work. And, as I will argue in a minute, my demonstrations are easier technically. If I wrote them down well enough, you may come to agree.

Spis treści

1. Restricting the Range.- 1 Summary of chapters 1 and 2.- 2. Inequalities.- 3. Standard transitions.- 4. Recurrence.- 5. Restricting the range.- 6. The Markov property.- 7. The convergence of XJ to X.- 8. The distribution of ?J given XJ.- 9. The joint distribution of .- 10. The convergence of QJ to Q.- 11. The distribution of X given XJ.- 2. Restricting the Range; Applications.- 1. Foreword.- 2. The generator.- 3. A theorem of Lévy.- 4. Determining the time scale.- 5. A theorem of Williams.- 6. Transformation of time.- 7. The transient case.- 3. Constructing the General Markov Chain.- 1. Introduction.- 2. The construction.- 3. A process with all states instantaneous and no pseudo jumps.- 4. An example of Kolmogorov.- 5. Slow convergence.- 6. Smith’s phenomenon.- 4. Append.- 1. Notation.- 2. Numbering.- 3. Bibliography.- Symbol Finder.