Marius Iosifescu - Böcker

Dependence with complete connections is a more general type of stochastic process than the well-known Markovian dependence, accounting for a complete history of a stochastic evolution. This book is an authoritative survey of knowledge of the subject, dealing with the basic theoretical understanding and also with applications. These arise in a variety of situations as diverse as stochastic models of learning, branching processes in random environments, continued fractions and dynamical systems. Thus the book will appeal to mathematicians working in probability theory, ergodic theory and number theory, as well as applied mathematicians, engineers, biologists and social scientists interested in applications of stochastic methods.

This book provides a pedagogical examination of the way in which stochastic models are encountered in applied sciences and techniques such as physics, engineering, biology and genetics, economics and social sciences. It covers Markov and semi-Markov models, as well as their particular cases: Poisson, renewal processes, branching processes, Ehrenfest models, genetic models, optimal stopping, reliability, reservoir theory, storage models, and queuing systems. Given this comprehensive treatment of the subject, students and researchers in applied sciences, as well as anyone looking for an introduction to stochastic models, will find this title of invaluable use.

The aim of" the present monograph is two-fold: (a) to give a short account of the main results concerning the theory of random systems with complete connections, and (b) to describe the general learning model by means of random systems with complete connections. The notion of chain with complete connections has been introduced in probability theory by ONICESCU and MIHOC (1935a). These authors have set themselves the aim to define a very broad type of dependence which takes into account the whole history of the evolution and thus includes as a special case the Markovian one. In a sequel of papers of the period 1935-1937, ONICESCU and MIHOC developed the theory of these chains for the homogeneous case with a finite set of states from differ­ ent points of view: ergodic behaviour, associated chain, limit laws. These results led to a chapter devoted to these chains, inserted by ONI­ CESCU and MIHOC in their monograph published in 1937. Important contributions to the theory of chains with complete connections are due to DOEBLIN and FORTET and refer to the period 1937-1940. They consist in the approach of chains with an infinite history (the so-called chains of infinite order) and in the use of methods from functional analysis.

This volume is a revised and enlarged version of Chapters 1 and 2 of a book with the same title, published in Romanian in 1968. The revision resulted in a new book which has been divided into two parts because of the large amount of new material. The present part is intended to introduce mathematicians and biologists with a strong mathematical and probabilistic background to the study of stochastic processes. We hope some readers will be able to discover by themselves the new features of our treatment such as the inclusion of some unusual topics, the special attention paid to some usual topics, and the grouping of the material. We draw the reader's attention to the numbering, because there are structural differences between the two parts. In Part I there are Chapters, Sections, Subsections, Paragraphs and Subparagraphs. Thus the numbering a. b. c. d. e refers to Subparagraph e of Paragraph d of Subsection c of Section b of Chapter a. Definitions, theorems lemmas and propositions are numbered a. b. n, n = 1,2, . . . , where a indicates the chapter and b the section. In Part II there are Sections, Subsections, Paragraphs, and SUbparagraphs. Thus the numbering a. b. c. d refers to Subparagraph d of Paragraph c of Subsection b of Section a. Theorems and lemmas are numbered a. n, n = 1, 2, . . . , where a indicates the section.