Let (X t) t2N 0 be a 3-state Markov chain with transition probability matrix P and statespacef1;2;3g.Findamapping f: f1;2;3g!f1;2g andamatrixPsuchthat(f(X t)) t2N 0 isnotaMarkovchain. Course information, a blog, discussion and resources for a course of 12 lectures on Markov Chains to second year mathematicians at Cambridge in autumn 2012. Proof. There are many nice exercises, some notes on the history of probability,  and on pages 464-466 there is information about A. Frederick Mosteller, 50 Challenging Problems in Probability, with Solutions, 1987. Let be a nite set we call our state space, and consider a sequence of alvued random avriables: (X 0;X 1;:::). When you comment, you can do this anonymously if you wish. They include: The notes have hyperlinks. Recurrence and transience; equivalence of transience and summability of n-step transition probabilities; equivalence of recurrence and certainty of return. Simple random walks in dimensions one, two and three. [1], 1 Definitions, basic properties, the transition matrix, 1.1 An example and some interesting questions, 3.1 Absorption probabilities and mean hitting times, 3.2 Calculation of hitting probabilities and mean hitting times, 3.3 Absorption probabilities are minimal solutions to RHEs, 4.1 Survival probability for birth death chains, 4.2 Mean hitting times are minimal solutions to RHEs, 5.2 Equivalence of recurrence and certainty of return, 5.3 Equivalence of transience and summability of n-step transition probabilities, 6.4 *A continuized analysis of random walk on Z^3*. This book it is particulary interesting about absorbing chains and mean passage times. This book is also quite easy to read. Grimmett and D.R. The extra questions are interesting and off the well-beaten path of questions that are typical for an introductory Markov Chains course. Peter Winkler, Mathematical Mind Benders, 2007. CUP 1997 (Chapter 1, Discrete Markov Chains is freely available to download. (We mention only a few names here; see the chapter Notes for references.) 1 Basics De nition 1. One emphasises probabilistic methods (as does Norris's book and our course); another is more matrix-based, as it this book. 2 ERRATA FOR MARKOV CHAINS AND MIXING TIMES and Corollary 1.17 implies ˇ z does not depend on z. This year I am organising these pages so that students can comment or ask questions on the discussion page, and also at the end of each of the blog posts. Here also are the overhead slides that I sometimes used in lectures. Long-run proportion of time spent in given state. Many of the puzzles are based in probability. 7.4 Invariant distribution is the solution to LHEs, 8.1 Existence and uniqueness up to constant multiples, 8.2 Mean return time, positive and null recurrence, 9.1 Equivalence of positive recurrence and the existence of an invariant distribution, 9.3 Convergence to equilibrium *and proof by coupling*, 10.2 *Kemeny's constant and the random target lemma*, 12.1 Reversibility and Ehrenfest's urn model, 12.4 *Random walks and electrical networks*, Reversible Markov Chains and Random Walks on Graphs, The PageRank citation ranking: bringing order to the web, non-technical books related to probability, Presenting probability via math puzzles is harmful, Appendix C. The probabilistic abacus for absorbing Markov chains, G.R. 2010 Mathematics Subject Classification. Definition and basic properties, the transition matrix. ), or answer an interesting question (that perhaps a student sends to me in email). This is not a book on Markov Chains, but a collection of mathematical puzzles that I recommend. You could read this (easy to understand) paper to learn more about the interesting connection between recurrence/transience properties of random walks and resistence in electrical network, as I will briefly discuss in Lecture 12. This is a book you wiill want to read if ever go beyond undergraduate study in this field. For statistical physicists Markov chains become useful in Monte Carlo simu- For example, Chapter 5 on Coupling, will tell you how the ideas we used in Lecture 9 can be extended. If you need to brush up of your knowledge of how to solve linear recurrence relations, see Section 1.11. The authors have good insight and you will find some gems here. A web page for the 2011 course home page is also still available. In other chapters this book provides a gentle introduction to probability and measure theory.) Chapters 1 and 2 nicely summarise most of the content of our course in a small number of pages. This will be sent to my email anonymously. [5] Chapters 1 and 2 nicely summarise most of the content of our course in a small number of pages. The modern theory of Markov chain mixing is the result of the convergence, in the 1980’s and 1990’s, of several threads. Markov Chains and Mixing Times Second Edition David A. Levin University of Oregon Yuval Peres Microsoft Research With contributions by Elizabeth L. Wilmer With a chapter on “Coupling from the Past” by James G. Propp and David B. Wilson AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 10.1090/mbk/107 . After reading the responses, I will forward them to the Faculty Office. The author does a good job of making difficult concepts seem fairly simple.) Stopping times and statement of the strong Markov property. [3] 2.2. For statistical physicists Markov chains become useful in M onte Carlo simu- This material is provided for students, supervisors (and others) to freely use in connection with this course. Usually they are deflned to have also discrete time (but deflnitions vary slightly in textbooks). I expect some good contributions. However, I reference this textbook mainly because it is a good place to read about some of the fascinating topics within the field of Markov chains that interest researchers today. The modern theory of Markov chain mixing is the result of the convergence, in the 1980’s and 1990’s, of several threads. I highly recommend that you read this chapter. the mixing time grows as the size of the state space increases. Notes on Markov Mixing Times by Peres et al. Each lecture has notes of 3.5–4 pages. Convergence to equilibrium for irreducible, positive recurrent, aperiodic chains *and proof by coupling*. It includes the "Evening out the Gumdrops" puzzle that I discuss in lectures, and lots of other great problems. There is a course blog in which I am writing a few comments after each lecture: to emphasize an idea, give a sidebar, correction (! The modern theory of Markov chain mixing is the result of the convergence, in the 1980’s and 1990’s, of several threads. The course closely follows Chapter 1 of James Norris's book, Markov Chains, 1998 (Chapter 1, Discrete Markov Chains is freely available to download and I recommend that you read it.) If you think you find a mistake in these notes, check that you have the most recent copy (as I may have already made a correction.) David Levin, Yuval Peres and Elizabeth Wilmer Markov Chains and Mixing Times, 2008. The probabilistic methods are more satisfying, but it is good to know something about the matrix methods too.) Mean return time, positive recurrence; equivalence of positive recurrence and the existence of an invariant distribution. Calculation of n-step transition probabilities. You should receive a supervision on each examples sheet. ), J.R. Norris Markov Chains. There are 2 examples sheets, each containing 13 questions, as well as 3 or 4 "extra" optional questions. If you click on an entry in the table of contents, or on a page number in the Index, you be taken to the appropriate page. This is discussed in brief in Lecture 8, Example 8.6 (random surfer). Invariant distributions, statement of existence and uniqueness up to constant multiples. Proposition 1.14 implies that ˇ z is a stationary distribution, 1. There are two distinct approaches to the study of Markov chains. Here is single fille of all the tripos examination questions on Markov Chains from 2001 to last June. OUP 2001 (Chapter 6.1-6.5 is on discrete Markov chains. Sheldon Ross, Introduction to Probability Models, 2006 (Chapter 4) and Stochastic Processes,1995 (Chapter 4) (Each of these books contains a readable chapter on Markov chains and many nice examples. However, I reference this textbook mainly because it is a good place to read about some of the fascinating topics within the field of Markov chains that interest researchers today.

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