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¡¡¡¡ In the Preface to the first edition, originally published in 1980, we mentioned that this book was based on the author's lectures in the Department of Mechanics and Mathematics of the Lomonosov University in Moscow, which were issued, in part, in mimeographed form under the title "Probability, Statistics, and Stochastic Processors, I, II" and published by that University. Our original intention in writing the first edition of this book was to divide the contents into three parts: probability, mathematical statistics, and theory of stochastic processes, which corresponds to an outline of a threesemester course of lectures for university students of mathematics. However, in the course of preparing the book, it turned out to be impossible to realize this intention completely, since a full exposition would have required too much space. In this connection, we stated in the Preface to the first edition that only probability theory and the theory of random processes with discrete time were really adequately presented.
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preface to the second edition
preface to the first edition
introduction
chapter i elementary probability theory
1. probabilistic model of an experiment with a finite number of outcomes
2. some classical models and distributions
3. conditional probability. independence
4. random variables and their properties
5. the bernoulli scheme. i. the law of large numbers
6. the bernoulli scheme. ii. limit theorems (local, de moivre-laplace, poisson)
7. estimating the probability of success in the bernoulli scheme
8. conditional probabilities and mathematical expectations with respect to decompositions
9. random walk. i. probabilities of ruin and mean duration in coin tossing
10. random walk. ii. reflection principle. arcsine law
11. martingales. some applications to the random walk
12. markov chains. ergodic theorem. strong markov property
chapter ii mathematical foundations of probability theory
1. probabilistic model for an experiment with infinitely many outcomes. kolmogorov's axioms
2. algebras and a-algebras. measurable spaces
3. methods of introducing probability measures on measurable spaces

¡¡¡¡Preface to the Second Edition
¡¡¡¡ In the Preface to the first edition, originally published in 1980, we mentioned that this book was based on the author's lectures in the Department of Mechanics and Mathematics of the Lomonosov University in Moscow, which were issued, in part, in mimeographed form under the title "Probability, Statistics, and Stochastic Processors, I, II" and published by that University. Our original intention in writing the first edition of this book was to divide the contents into three parts: probability, mathematical statistics, and theory of stochastic processes, which corresponds to an outline of a threesemester course of lectures for university students of mathematics. However, in the course of preparing the book, it turned out to be impossible to realize this intention completely, since a full exposition would have required too much space. In this connection, we stated in the Preface to the first edition that only probability theory and the theory of random processes with discrete time were really adequately presented.
¡¡¡¡ Essentially all of the first edition is reproduced in this second edition. Changes and corrections are, as a rule, editorial, taking into account comments made by both Russian and foreign readers of the Russian original and of the English and German translations IS11']. The author is grateful to all of these readers for their attention, advice, and helpful criticisms.
¡¡¡¡ In this second English edition, new material also has been added, as follows: in Chapter III, 5, 7-12; in Chapter IV, 5; in Chapter VII, 8-10. The most important addition is the third chapter. There the reader will find expositions of a number of problems connected with a deeper study of themes such as the distance between probability measures, metrization of weak convergence, and contiguity of probability measures. In the same chapter, we have added proofs of a number of important results on the rapidity of convergence in the central limit theorem and in Poisson's theorem on the approximation of the binomial by the Poisson distribution. These were merely stated in the first edition.
¡¡¡¡ We also call attention to the new material on the probability of large deviations (Chapter IV, 5), on the central limit theorem for sums of dependent random variables (Chapter VII, 8), and on 9 and 10 of Chapter VII.
¡¡¡¡ During the last few years, the literature on probability published in Russia by Nauka has been extended by Sevastyanov [S10], 1982; Rozanov [R6], 1985; Borovkov [B4], 1986; and Gnedenko [G4], 1988. It appears that these publications, together with the present volume, being quite different and complementing each other, cover an extensive amount of material that is essentially broad enough to satisfy contemporary demands by students in various branches of mathematics and physics for instruction in topics in probability theory.
¡¡¡¡ Gnedenko's textbook [G4] contains many well-chosen examples, includ ing applications, together with pedagogical material and extensive surveys of the history of probability theory. Borovkov's textbook I-B4] is perhaps the ,most like the present book in the style of exposition. Chapters 9 (Elements of Renewal Theory), 11 (Factorization of the Identity) and 17 (Functional Limit Theorems), which distinguish [B4] from this book and from [G4] and IR6], deserve special mention. Rozanov's textbook contains a great deal of mate rial on a variety of mathematical models which the theory of probability and mathematical statistics provides for describing random phenomena and their evolution. The textbook by Sevastyanov is based on his two-semester course at the Moscow State University. The material in its last four chapters covers the minimum amount of probability and mathematical statistics required in a one-year university program. In our text, perhaps to a greater extent than in those mentioned above, a significant amount of space is given to settheoretic aspects and mathematical foundations of probability theory.
¡¡¡¡ Exercises and problems are given in the books by Gnedenko and Sevastyanov at the ends of chapters, and in the present textbook at the end of each section. These, together with, for example, the problem sets by A. V. Prokhorov and V. G. and N. G. Ushakov (Problems in Probability Theory, Nauka, Moscow, 1986) and by Zubkov, Sevastyanov, and Chistyakov (Col lected Problems in Probability Theory, Nauka, Moscow, 1988) can be used by readers for independent study, and by teachers as a basis for seminars for students.
¡¡¡¡ Special thanks to Harold Boas, who kindly translated the revisions from Russian to English for this new edition.
¡¡¡¡ Moscow
¡¡¡¡ A. Shiryaev
¡¡¡¡ Preface to the First Edition
¡¡¡¡ This textbook is based on a three-semester course of lectures given by the author in recent years in the Mechanics-Mathematics Faculty of Moscow State University and issued, in part, in mimeographed form under the title Probability, Statistics, Stochastic Processes, I, H by the Moscow State University Press.
¡¡¡¡ We follow tradition by devoting the first part of the course (roughly one semester) to the elementary theory of probability (Chapter I). This begins with the construction of probabilistic models with finitely many outcomes and introduces such fundamental probabilistic concepts as sample spaces, events, probability, independence, random variables, expectation, correlation, conditional probabilities, and so on.
¡¡¡¡ Many probabilistic and statistical regularities are effectively illustrated even by the simplest random walk generated by Bernoulli trials. In this connection we study both classical results (law of large numbers, local and integral De Moivre and Laplace theorems) and more modern results (for example, the arc sine law).
¡¡¡¡ The first chapter concludes with a discussion of dependent random variables generated by martingales and by Markov chains.
¡¡¡¡ Chapters II-IV form an expanded version of the second part of the course (second semester). Here we present (Chapter II) Kolmogorov's generally accepted axiomatization of probability theory and the mathematical methods that constitute the tools of modern probability theory (a-algebras, measures and their representations, the Lebesgue integral, random variables and random elements, characteristic functions, conditional expectation with respect to a a-algebra, Gaussian systems, and so on). Note that two measuretheoretical results--Caratheodory's theorem on the extension of measures and the Radon-Nikodym theorem--are quoted without proof.
¡¡¡¡ The third chapter is devoted to problems about weak convergence of probability distributions and the method of characteristic functions for proving limit theorems. We introduce the concepts of relative compactness and tightness of families of probability distributions, and l/rove (for the real line) Prohorov's theorem on the equivalence of these concepts.
¡¡¡¡ The same part of the course discusses properties "with probability 1" for sequences and sums of independent random variables (Chapter IV). We give proofs of the "zero or one laws" of Kolmogorov and of Hewitt and Savage, tests for the convergence of series, and conditions for the strong law of large numbers. The law of the iterated logarithm is stated for arbitrary sequences of independent identically distributed random variables with finite second moments, and proved under the assumption that the variables have Gaussian distributions.
¡¡¡¡ Finally, the third part of the book (Chapters V-VIII) is devoted to random processes with discrete parameters (random sequences). Chapters V and VI are devoted to the theory of stationary random sequences, where "stationary'' is interpreted either in the strict or the wide sense. The theory of random sequences that are stationary in the strict sense is based on the ideas of ergodic theory: measure preserving transformations, ergodicity, mixing, etc. We reproduce a simple proof (by A. Garsia) of the maximal ergodic theorem; this also lets us give a simple proof of the Birkhoff-Khinchin ergodic theorem.