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Reversible Markov Chains on Trees

Joshua Walter Lovelace

出版	California State University San Marcos, 2009
URL	http://books.google.com.hk/books?id=MdRWQwAACAAJ&hl=&source=gbs_api

註釋

This thesis will briefly go over definitions and properties of continuous time Markov chains and describe the definition of reversibility and conditions for positive recurrence. We then discuss the evolution of the Markov chains studied in [12] which have state spaces consisting of the finite rooted subtrees of the infinite, regular, rooted, positional, d-ary tree. We summarize how the author of [12] used reversibility to prove a positive recurrence result for these chains. For the main results of the thesis, we analyze a Markov chain with a state space consisting of the set of finite rooted trees with unbounded degree. This chain was proposed and analyzed in [4] without reference to reversibility. Here, we identify a reversible measure for that chain and show nnder what conditions the measure is summable, and therefore give an alternate proof of the positive recurrence result in [4] using more probabilistic methods. Finally, the reversible measure is used to discuss the behavior of the chain, when it is positive recurrent.