Markov Chains with Stationary Transition Probabilities

ISBN-13: 9783642494086 / Angielski / Miękka / 1960 / 278 str.

Kai Lai Chung; R. Grammel; F. Hirzebruch

Opis

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."

Spis treści

I. Discrete Parameter.- § 1. Fundamental definitions.- § 2. Transition probabilities.- § 3. Classification of states.- § 4. Recurrence.- § 5. Criteria and examples.- § 6. The main limit theorem.- § 7. Various complements.- § 8. Repetitive pattern and renewal process.- § 9. Taboo probabilities.- § 10. The generating function.- § 11. The moments of first entrance time distributions.- § 12. A random walk example.- § 13. System theorems.- § 14. Functionals and associated random variables.- § 15. Ergodic theorems.- § 16. Further limit theorems.- § 17. Almost closed and sojourn sets.- II. Continuous Parameter.- § 1. Transition matrix: basic properties.- § 2. Standard transition matrix.- § 3. Differentiability.- § 4. Definitions and measure-theoretic foundations.- § 5. The sets of constancy.- § 6. Continuity properties of sample functions.- § 7. Further specifications of the process.- § 8. Optional random variable.- § 9. Strong Markov property.- § 10. Classification of states.- § 11. Taboo probability functions.- § 12. Ratio limit theorems.- § 13. Discrete approximations.- § 14. Functionals.- § 15. Post-exit process.- § 16. Imbedded renewal process.- § 17. The two systems of differential equations.- § 18. The minimal solution.- § 19. The first infinity.- § 20 Examples.- Addenda.

O autorze

Bartel van der Waerden, geb. am 2.2.1903 in Amsterdam, ging 1924 ging als Student nach Göttingen und wurde dort mit Emmy Noether und der abstrakten Algebra bekannt. Sein Hauptinteresse galt damals vor allem der Begründung der algebraischen Geometrie mit Hilfe der neuen algebraischen Methoden. Als er im Jahre 1926 als junger Doktor mit einem Rockefeller-Stipendium nach Hamburg kam, hatte er Gelegenheit, eine didaktisch hervorragende Algebra-Vorlesung von Emil Artin zu hören. Die Ausarbeitung, die er von dieser Vorlesung machte, wurde zum Kern des vorliegenden Werkes. Es erschien zuerst 1930-31 unter dem Titel 'Moderne Algebra' in der Sammlung 'Grundlehren der mathematischen Wissenschaften'. In der Folge wurde das Werk in die englische, russische und chinesische Sprache übersetzt. Im Jahre 1928 wurde der Autor Professor an der Universität Groningen. Seit 1951 lebte und arbeitete er bis zu seiner Emeritierung in Zürich als Professor an der dortigen Universität.