Markov Chains

ISBN-13: 9781461255024 / Angielski / Miękka / 2011 / 382 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 the third. This last book is a monograph which explains one way to think about chains with instantaneous states. The results in it are supposed to be new, except where there are specific disclaim ers; it's written in the framework of Markov Chains. 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 VB1 PREFACE argue in a minute, my demonstrations are easier technically. If I wrote them down well enough, you may come to agree."

Spis treści

I. Discrete time.- 1. Introduction to Discrete Time.- 1. Foreword.- 2. Summary.- 3. The Markov and strong Markov properties.- 4. Classification of states.- 5. Recurrence.- 6. The renewal theorem.- 7. The limits of Pn.- 8. Positive recurrence.- 9. Invariant probabilities.- 10. The Bernoulli walk.- 11. Forbidden transitions.- 12. The Harris walk.- 13. The tail ?-field and a theorem of Orey.- 14. Examples.- 2. Ratio Limit Theorems.- 1. Introduction.- 2. Reversal of time.- 3. Proofs of Derman and Doeblin.- 4. Variations.- 5. Restricting the range.- 6. Proof of Kingman-Orey.- 7. An example of Dyson.- 8. Almost everywhere ratio limit theorems.- 9. The sum of a function over different j-blocks.- 3. Some Invariance Principles.- 1. Introduction.- 2. Estimating the partial sums.- 3. The number of positive sums.- 4. Some invariance principles.- 5. The concentration function.- 4. The Boundary.- 1. Introduction.- 2. Proofs.- 3. A convergence theorem.- 4. Examples.- 5. The last visit to i before the first visit to J\.- II. Continuous time.- 5. Introduction to Continuous Time.- 1. Semigroups and processes.- 2. Analytic properties.- 3. Uniform semigroups.- 4. Uniform substochastic semigroups.- 5. The exponential distribution.- 6. The step function case.- 7. The uniform case.- 6. Examples for the Stable Case.- 1. Introduction.- 2. The first construction.- 3. Examples on the first construction.- 4. The second construction.- 5. Examples on the second construction.- 6. Markov times.- 7. Crossing the infinities.- 7. The Stable Case.- 1. Introduction.- 2. Regular sample functions.- 3. The post-exit process.- 4. The strong Markov property.- 5. The minimal solution.- 6. The backward and forward equations.- 8. More Examples for the Stable Case.- 1. An oscillating semigroup.- 2. A semigroup with an infinite second derivative.- 3. Large oscillations in P(t, 1, 1).- 4. An example of Speakman.- 5. The embedded jump process is not Markov.- 6. Isolated infinities.- 7. The set of infinities is bad.- 9. The General Case.- 1. An example of Blackwell.- 2. Quasiregular sample functions.- 3. The sets of constancy.- 4. The strong Markov property.- 5. The post-exit process.- 6. The abstract case.- III..- 10. Appendix.- 1. Notation.- 2. Numbering.- 3. Bibliography.- 4. The abstract Lebesgue integral.- 5. Atoms.- 6. Independence.- 7. Conditioning.- 8. Martingales.- 9. Metric spaces.- 10. Regular conditional distributions.- 11. The Kolmogorov consistency theorem.- 12. The diagonal argument.- 13. Classical Lebesgue measure.- 14. Real variables.- 15. Absolute continuity.- 16. Convex functions.- 17. Complex variables.- Symbol Finder.