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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.
Reviewer: H.Daduna

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60Gxx Stochastic processes
60Jxx Markov processes