02987nam a22003375a 4500001001100000003001200011005001700023006001900040007001500059008004100074020001800115024002100133040001400154072001600168084001500184100003000199245013500229260008200364300003400446336002600480337002600506338003600532347002400568490003900592506006500631520176700696650003702463650005302500856003202553856006402585104-091109CH-001817-320091109150325.0a fot ||| 0|cr nn mmmmamaa091109e20090815sz fot ||| 0|eng d a978303719571070a10.4171/0712doi ach0018173 7aPBT2bicssc a60-xx2msc1 aWoess, Wolfgang,eauthor.10aDenumerable Markov Chainsh[electronic resource] :bGenerating Functions, Boundary Theory, Random Walks on Trees /cWolfgang Woess3 aZuerich, Switzerland :bEuropean Mathematical Society Publishing House,c2009 a1 online resource (368 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aEMS Textbooks in Mathematics (ETB)1 aRestricted to subscribers:uhttp://www.ems-ph.org/ebooks.php 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.07aProbability & statistics2bicssc07aProbability theory and stochastic processes2msc40uhttps://doi.org/10.4171/071423cover imageuhttp://www.ems-ph.org/img/books/woess_mini.jpg