General irreducible Markov chains and non-negative operators.

*(English)*Zbl 0551.60066
Cambridge Tracts in Mathematics, 83. Cambridge etc.: Cambridge University Press. XI, 156 p. £20.00; $ 37.50 (1984).

The book follows the terminology calling Markov chain a Markov process in discrete time but with arbitrary state space. Essentially only irreducible chains are discussed, i.e. chains where there exists a \(\sigma\)-finite measure \(\phi\) on the state space such that from any starting point each set of positive \(\phi\)-measure can be reached with positive probability. This leaves aside most infinite particle models.

The theory of such irreducible chains has profited much in recent years by a socalled splitting technique invented by the author [Z. Wahrscheinlichkeitstheor. Verw. Geb. 43, 309-318 (1978; Zbl 0364.60104)] and independently by K. B. Athreya and P. Ney [Trans. Am. Math. Soc. 245, 493-501 (1978; Zbl 0397.60053)].

This technique is based on the introduction of atoms. An atom for a Markov transition kernel P is a measurable subset A of the state space such that P(x,\(\cdot)\) is constant on \(x\in A\). If a kernel has such an atom of positive \(\phi\)-measure, one can consider the successive entrance times in this atom, cutting in this way the chain into independent parts. This technique is well known for chains with a countable state space where one simply takes a suitable point as an atom, and it has proved to be very useful there. For a chain with general state space such atoms usually do not exist. However, in an ingenious way, the author succeeded to enlarge (or split) the state space in such a way that an atom appears (at least for some iterate of P). This construction simplifies much of the theory of Markov chains, e.g. convergence of iterates of P, R- recurrence, central limit theorems etc.

The book develops this approach in a very nice and well readable way.

The theory of such irreducible chains has profited much in recent years by a socalled splitting technique invented by the author [Z. Wahrscheinlichkeitstheor. Verw. Geb. 43, 309-318 (1978; Zbl 0364.60104)] and independently by K. B. Athreya and P. Ney [Trans. Am. Math. Soc. 245, 493-501 (1978; Zbl 0397.60053)].

This technique is based on the introduction of atoms. An atom for a Markov transition kernel P is a measurable subset A of the state space such that P(x,\(\cdot)\) is constant on \(x\in A\). If a kernel has such an atom of positive \(\phi\)-measure, one can consider the successive entrance times in this atom, cutting in this way the chain into independent parts. This technique is well known for chains with a countable state space where one simply takes a suitable point as an atom, and it has proved to be very useful there. For a chain with general state space such atoms usually do not exist. However, in an ingenious way, the author succeeded to enlarge (or split) the state space in such a way that an atom appears (at least for some iterate of P). This construction simplifies much of the theory of Markov chains, e.g. convergence of iterates of P, R- recurrence, central limit theorems etc.

The book develops this approach in a very nice and well readable way.

Reviewer: E.Bolthausen