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Markov Chains with Stationary Transition Probabilities (Paperback, Softcover reprint of the original 1st ed. 1960)
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Markov Chains with Stationary Transition Probabilities (Paperback, Softcover reprint of the original 1st ed. 1960)
Series: Grundlehren der mathematischen Wissenschaften, 104
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The theory of Markov chains, although a special case of Markov
processes, is here developed for its own sake and presented on its
own merits. In general, the hypothesis of a denumerable state
space, which is the defining hypothesis of what we call a "chain"
here, generates more clear-cut questions and demands more precise
and definitive an swers. For example, the principal limit theorem (
1. 6, II. 10), still the object of research for general Markov
processes, is here in its neat final form; and the strong Markov
property ( 11. 9) is here always applicable. While probability
theory has advanced far enough that a degree of sophistication is
needed even in the limited context of this book, it is still
possible here to keep the proportion of definitions to theorems
relatively low. . From the standpoint of the general theory of
stochastic processes, a continuous parameter Markov chain appears
to be the first essentially discontinuous process that has been
studied in some detail. It is common that the sample functions of
such a chain have discontinuities worse than jumps, and these baser
discontinuities play a central role in the theory, of which the
mystery remains to be completely unraveled. In this connection the
basic concepts of separability and measurability, which are usually
applied only at an early stage of the discussion to establish a
certain smoothness of the sample functions, are here applied
constantly as indispensable tools.
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