Markov chains. 2nd rev. ed.

*(English)*Zbl 0539.60073
North-Holland Mathematical Library, Vol. 11. Amsterdam - New York - Oxford: North-Holland. XI, 374 p. $ 57.75; Dfl. 150.00 (1984).

The book under review is the second (revised) edition of the 1975 version (see Zbl 0332.60045). As clearly stated in the preface to the second edition this differs from the previous one by the addition of a few new topics, among which the most important are the subadditive ergodic theory and the ”zero-two” laws.

The main results on subadditive ergodic theory (Ch. 4, §6) are taken from the work of Kingman. The decomposition theorem as well as Kingman’s convergence theorem are proved and several applications are given at the end of the section. Both ”zero-two” laws due to Ornstein and Sucheston are presented in Ch. 6, §1 and subsequently used in Ch. 6, §2 to prove results on cyclic classes and limit theorems carried over to general Harris chains by Jamison and Orey.

A number of various exercises were added to each section, some of them summarizing important results obtained in the period between the two editions (see for instance Ex.2.23, Ch. 3, §2 providing an introduction to R-theory). As one would have expected the notes and comments as well as the list of references are brought up to date.

The main results on subadditive ergodic theory (Ch. 4, §6) are taken from the work of Kingman. The decomposition theorem as well as Kingman’s convergence theorem are proved and several applications are given at the end of the section. Both ”zero-two” laws due to Ornstein and Sucheston are presented in Ch. 6, §1 and subsequently used in Ch. 6, §2 to prove results on cyclic classes and limit theorems carried over to general Harris chains by Jamison and Orey.

A number of various exercises were added to each section, some of them summarizing important results obtained in the period between the two editions (see for instance Ex.2.23, Ch. 3, §2 providing an introduction to R-theory). As one would have expected the notes and comments as well as the list of references are brought up to date.

Reviewer: G.Oprisan

##### MSC:

60J27 | Continuous-time Markov processes on discrete state spaces |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |