LEADER 03025nam 22003855a 450 001 9910151933003321 005 20091109150325.0 010 $a3-03719-571-1 024 70$a10.4171/071 035 $a(CKB)3710000000953845 035 $a(CH-001817-3)104-091109 035 $a(PPN)178155659 035 $a(EXLCZ)993710000000953845 100 $a20091109j20090815 fy 0 101 0 $aeng 135 $aurnn|mmmmamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aDenumerable Markov Chains$b[electronic resource] $eGenerating Functions, Boundary Theory, Random Walks on Trees /$fWolfgang Woess 210 3 $aZuerich, Switzerland $cEuropean Mathematical Society Publishing House$d2009 215 $a1 online resource (368 pages) 225 0 $aEMS Textbooks in Mathematics (ETB) 330 $aMarkov chains are the first and most important examples of random processes. This book is about time-homogeneous Markov chains that evolve with discrete time steps on a countable state space. Measure theory is not avoided, careful and complete proofs are provided. A specific feature is the systematic use, on a relatively elementary level, of generating functions associated with transition probabilities for analyzing Markov chains. Basic definitions and facts include the construction of the trajectory space and are followed by ample material concerning recurrence and transience, the convergence and ergodic theorems for positive recurrent chains. There is a side-trip to the Perron-Frobenius theorem. Special attention is given to reversible Markov chains and to basic mathematical models of "population evolution" such as birth-and-death chains, Galton-Watson process and branching Markov chains. A good part of the second half is devoted to the introduction of the basic language and elements of the potential theory of transient Markov chains. Here the construction and properties of the Martin boundary for describing positive harmonic functions are crucial. In the long final chapter on nearest neighbour random walks on (typically infinite) trees the reader can harvest from the seed of methods laid out so far, in order to obtain a rather detailed understanding of a specific, broad class of Markov chains. The level varies from basic to more advanced, addressing an audience from master's degree students to researchers in mathematics, and persons who want to teach the subject on a medium or advanced level. A specific characteristic of the book is the rich source of classroom-tested exercises with solutions. 606 $aProbability & statistics$2bicssc 606 $aProbability theory and stochastic processes$2msc 615 07$aProbability & statistics 615 07$aProbability theory and stochastic processes 686 $a60-xx$2msc 700 $aWoess$b Wolfgang$060805 801 0$bch0018173 906 $aBOOK 912 $a9910151933003321 996 $aDenumerable Markov Chains$92567447 997 $aUNINA