Approximating Countable Markov ChainsThe book, part of a trilogy covering the field of Markov processes, explains one method of approximating countable Markov chains by finite ones. Intended for use in seminars with advanced graduate students, it is written in the framework of the first book in the trilogy, Markov Chains, although it is completely independent of the second, Brownian Motion and Diffusion. The idea is to skip over the times at which the chain is outside some large, finite set of states. The technique is especially useful for dealing with instantaneous states. Many of the results are original. (Author). |
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Contents
RESTRICTING THE RANGE | 1 |
INTRODUCTION TO DISCRETE TIME | 4 |
RESTRICTING THE RANGE APPLICATIONS | 64 |
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a₁ Abbreviate absorbing argument B₁ binary rational Borel chapter Claim coincides construction converges countably infinite defined empty product exponentially distributed F(1)-measurable Figure finite subset Fubini given hitting holding I-valued i₁ implies In+1 independent and exponentially integers interval of constancy joint distribution jump least Lebesgue measure Lemma Let F locally finitary Markov chain Markov property Markov with stationary Markov with transitions martingale matrix nondecreasing notation null set o-field spanned P-independent P-probability P(do P₁ Poisson process Poisson with parameter positive recurrent Prob probability triple prove pseudo-jumps q-lim quasiregular random variable restriction retracted right continuous sample functions satisfies Section sequence standard stochastic semigroup starting stationary transitions step functions stochastic matrix strictly increasing strong Markov property substochastic T₁ Theorem TJ,n visits X-scale X-time X₁ XN+1 Y₁ Z₁