Sinal是统计物理学的数学问题的国际权威，特别是对遍历理论有突出贡献．他首次给出任意保测映射的不变量摘以严格定义．其后证明弹子运动的遍历性以及拟周期Schr?dinger方程的谱性质·



Sinai,Yakov （1935.-）
俄国数学家，生于莫斯科．1953年进入莫斯科大学学习，1957年毕业．1960年获副博士学位，1963年获博士学位，其后留校研究，1971年升任教授．1971年起任苏联科学院Landau理论物理研究所研究员．1993年起兼任美国普林斯顿大学教授．
他是俄国科学院院士及美国文理科学院院士，他还是伦敦数学会名誉会员．他曾荣获Boltzmann金质奖章（1986）；Heinemann奖（1989）；Markov奖（1990）以及意大利Trieste国际理论物理中心Dirac奖章（1992）．1997年因“对统计力学中数学严格方法及动力系统的遍历理论以及它们在物理学中的应用所做的基本贡献”而获Wolf奖.
Probability Theory- An  Introductory  Course
Translated  from  the  Russian
by  D.  Haughton




Leonid B. Koralov Yakov G. Sinai
Theory of Probability  and Random Processes



Probability Theory -An  Introductory  Course

Prefaces
Preface  to  the  English Edition
This book grew  out  of lecture courses  that  were given  to  second and third
year students  at  the Mathematics Department of  Moscow  State University for
many years.  It  requires some knowledge of measure theory, and the modern
language of the theory of measurable partitions  is  frequently used.  In  several
places I have tried  to  emphasize some of the connections with current trends
in probability theory and statistical mechanics.
I  thank Professor Haughton for her excellent translation.
Ya.  G. Sinai  May  1992
Preface  to  First Ten Chapters
The first ten chapters of this book comprise an extended version of the first
part  of a required course in Probability Theory, which  I  have been teaching for
may years  to  fourth semester students in the Department of Mechanics  and
Mathematics  at  Moscow State University. The fundamental idea  is  to  give a
logically coherent introduction  to  the subject, while making as little use as
possible of the apparatus of measure theory and Lebesgue integration. To this
end, it was necessary  to  modify a number of details in the presentation of
some long-established sections.
These chapters cover the concepts of random variable, mathematical expec-
tation  and  variance, as well as sequences of independent trials, Markov chains,
and random walks on a lattice. Kolmogorov's axioms are used throughout the
text. Several non-traditional topics  are  also touched upon, including the prob-
lem of percolation, and the introduction of conditional probability through
the concept of measurable partitions. With the inclusion of these topics it  is
hoped  that  students will become actively involved in scientific research  at  an
early stage.
This  part  of the book was refereed by B.V. Gnedenko, Member of the
Academy of Science of the Ukrainian Socialist Soviet Republic,  and  N.N.
Chentsov, Doctor of Mathematical and Physical Sciences. I wish  to  thank
I.S. Sineva for assistance in preparing the original manuscript for publication.
Preface  to  Chapters  11  - 16
Chapters eleven through sixteen constitute the second  part  of a course on
Probability Theory for mathematics students in the Department of Mechanics
and Mathematics of Moscow State University. The chapters cover  the  strong
law  of large numbers, the weak convergence of probability distributions, and
the central limit theorem for sums of independent random variables. The no-
tion of stability, as it relates to the central limit theorem,  is  discussed from the
point of view of the method of renormalization group theory in statistical
physics, as is a somewhat less traditional topic, as  is  the analysis of asymptotic
probabilities of large deviations.
This  part  of the book was also refereed by B.V. Gnedenko, as well as
by  Professor A.N. Shiryaev. I wish to thank M.L. Blank,  A.  Dzhalilov, E.O.
Lokutsievckaya and I.S. Sineva for their great assistance in preparing  the  orig-
inal manuscript for publication.
Translator's  Preface
The Russian version of this book  was  published in two parts.  Part  I, covering
the first ten chapters, appeared in 1985.  Part  II appeared in 1986, and only
the first six chapters have been translated here.
I would like to thank Jonathan Haughton for typesetting the English ver-
sion in  TEX.

Leonid B. Koralov Yakov G. Sinai
Theory of Probability  and Random Processes
Second Edition
Preface
This book is primarily based on a one-year course that has been taught for
a number of years at Princeton University to advanced undergraduate and
graduate students. During the last year a similar course has also been taught
at the University of Maryland.
We would like to express our thanks to Ms. Sophie Lucas and Prof. Rafael
Herrera who read the manuscript and suggested many corrections. We are
particularly grateful to Prof. Boris Gurevich for making many important sug-
gestions on both the mathematical content and style.
While writing this book, L. Koralov was supported by a National Sci-
ence Foundation grant (DMS-0405152). Y. Sinai was supported by a National
Science Foundation grant (DMS-0600996).
Leonid Koralov
Yakov Sinai
Part I Probability Theory
1 Random Variables and Their Distributions . . . . . . . . . . . . . . . . 3
1.1 Spaces of Elementary Outcomes, σ-Algebras, and Measures . . . 3
1.2 Expectation and Variance of Random Variables
on a Discrete Probability Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Probability of a Union of Events . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Equivalent Formulations of σ-Additivity, Borel σ-Algebras
and Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Distribution Functions and Densities . . . . . . . . . . . . . . . . . . . . . . . 19
1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Sequences of Independent Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Law of Large Numbers and Applications . . . . . . . . . . . . . . . . . . . 25
2.2 de Moivre-Laplace Limit Theorem and Applications . . . . . . . . . 32
2.3 Poisson Limit Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Lebesgue Integral and Mathematical Expectation. . . . . . . . . . 37
3.1 Definition of the Lebesgue Integral. . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Induced Measures and Distribution Functions . . . . . . . . . . . . . . . 41
3.3 Types of Measures and Distribution Functions . . . . . . . . . . . . . . 45
3.4 Remarks on the Construction of the Lebesgue Measure . . . . . . . 47
3.5 Convergence of Functions, Their Integrals, and the Fubini
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6 Signed Measures and the Radon-Nikodym Theorem . . . . . . . . . . 52
3.7 L p Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.8 Monte Carlo Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
VIII Contents
4 Conditional Probabilities and Independence . . . . . . . . . . . . . . . 59
4.1 Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Independence of Events, σ-Algebras, and Random Variables . . 60
4.3 π-Systems and Independence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Markov Chains with a Finite Number of States . . . . . . . . . . . . 67
5.1 Stochastic Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Ergodic and Non-Ergodic Markov Chains . . . . . . . . . . . . . . . . . . . 71
5.4 Law of Large Numbers and the Entropy of a Markov Chain . . . 74
5.5 Products of Positive Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.6 General Markov Chains and the Doeblin Condition . . . . . . . . . . 78
5.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6 Random Walks on the Lattice Z d . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1 Recurrent and Transient Random Walks. . . . . . . . . . . . . . . . . . . . 85
6.2 Random Walk on Z and the Reflection Principle. . . . . . . . . . . . . 88
6.3 Arcsine Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4 Gambler’s Ruin Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7 Laws of Large Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.1 Definitions, the Borel-Cantelli Lemmas, and the Kolmogorov
Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 Kolmogorov Theorems on the Strong Law of Large Numbers . . 103
7.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8 Weak Convergence of Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.1 Definition of Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2 Weak Convergence and Distribution Functions . . . . . . . . . . . . . . 111
8.3 Weak Compactness, Tightness, and the Prokhorov Theorem . . 113
8.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.2 Characteristic Functions and Weak Convergence. . . . . . . . . . . . . 123
9.3 Gaussian Random Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
10 Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
10.1 Central Limit Theorem, the Lindeberg Condition . . . . . . . . . . . . 131
10.2 Local Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10.3 Central Limit Theorem and Renormalization Group Theory . . 139
10.4 Probabilities of Large Deviations . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Contents IX
16.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
17 Generalized Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
17.1 Generalized Functions and Generalized Random Processes . . . . 247
17.2 Gaussian Processes and White Noise . . . . . . . . . . . . . . . . . . . . . . . 251
18 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
22.2 An Example of a Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . 346
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349