The theory of stochastic processes I. Transl. from the Russian by S. Kotz. Corr. printing of the 1st ed.

*(English)*Zbl 0531.60001
Grundlehren der mathematischen Wissenschaften, 210. Berlin, Heidelberg, New York: Springer-Verlag. VIII, 574 p. DM 148.00; $ 87.40 (1980).

For a review of the Russian edition see Zbl 0298.60023.

Chapt. I: A short review of probability theory. Chapt. II: Random sequences including martingales, Markov chains, ergodic theorems. Chapt. III: Random functions: Some important special classes, general theorems on separability, measurability, path properties. Chapt. IV: Linear theory of random processes: Correlation theory for wide-sense stationary processes, spectral representation for random sequences and random fields, Hilbert random functions, stochastic integrals and integral representation of random functions, forecasting and filtering. Chapt. V: Probability measures on functional spaces: measures on function spaces associated with random functions, measures on metric spaces, etc., Gauss measures on Hilbert spaces. Chapt. VI: Limit theorems for random processes: weak convergence in metric and Hilbert spaces, limit theorems for processes with continuous paths or with paths having only discontinuities of the first kind. Chapt. VII: Absolute continuity of measures associated with random processes: General theorems, admissible shifts in Hilbert spaces, Gauss measures, stationary Gauss processes, Markov processes. Chapt. VIII: Measurable functions on Hilbert spaces. The book is closed by a short section with historical and bibliographical remarks, a bibliography and corrections to the first printings of Vol. I, II.

Chapters II, III, IV are a comprehensive text on classical topics of the theory of stochastic processes on a mathematical rigorous level. Dealing only with the theoretical part of the field the authors are able to write down a very clear and streamlined text, see e.g. the parts on Markov chains or on separable, measurable processes. Chapters V-VIII present the modern theory of stochastic processes as far as they can be dealt with via measure theory on function spaces. As the necessity for a second printing indicates in the last ten years the book has become a wide- spread reference manual for stochastic processes - due to its clarity of exposition, its completeness, and due to the increasing importance of the field in theory and in practice.

Chapt. I: A short review of probability theory. Chapt. II: Random sequences including martingales, Markov chains, ergodic theorems. Chapt. III: Random functions: Some important special classes, general theorems on separability, measurability, path properties. Chapt. IV: Linear theory of random processes: Correlation theory for wide-sense stationary processes, spectral representation for random sequences and random fields, Hilbert random functions, stochastic integrals and integral representation of random functions, forecasting and filtering. Chapt. V: Probability measures on functional spaces: measures on function spaces associated with random functions, measures on metric spaces, etc., Gauss measures on Hilbert spaces. Chapt. VI: Limit theorems for random processes: weak convergence in metric and Hilbert spaces, limit theorems for processes with continuous paths or with paths having only discontinuities of the first kind. Chapt. VII: Absolute continuity of measures associated with random processes: General theorems, admissible shifts in Hilbert spaces, Gauss measures, stationary Gauss processes, Markov processes. Chapt. VIII: Measurable functions on Hilbert spaces. The book is closed by a short section with historical and bibliographical remarks, a bibliography and corrections to the first printings of Vol. I, II.

Chapters II, III, IV are a comprehensive text on classical topics of the theory of stochastic processes on a mathematical rigorous level. Dealing only with the theoretical part of the field the authors are able to write down a very clear and streamlined text, see e.g. the parts on Markov chains or on separable, measurable processes. Chapters V-VIII present the modern theory of stochastic processes as far as they can be dealt with via measure theory on function spaces. As the necessity for a second printing indicates in the last ten years the book has become a wide- spread reference manual for stochastic processes - due to its clarity of exposition, its completeness, and due to the increasing importance of the field in theory and in practice.

Reviewer: H.Daduna

##### MSC:

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

60Gxx | Stochastic processes |

60Jxx | Markov processes |