Introduction to probability theory / Kiyosi Itô.

Format:

Book

Author/Creator:

Itō, Kiyosi, 1915-2008.

Standardized Title:

Kakuritsuron. 1-4-shō. English

Language:

English

Japanese

Subjects (All):

Probabilities.

Physical Description:

ix, 213 pages ; 24 cm

Place of Publication:

Cambridge [Cambridgeshire] ; New York : Cambridge University Press, 1984.

Summary:

Professor Ito is one of the most distinguished probability theorists in the world, and in this modern, concise introduction to the subject he explains basic probabilistic concepts rigorously and yet gives at the same time an intuitive understanding of random phenomena.

In the first chapter he considers finite situations, but from an advanced standpoint that enables the transition to greater generality to be achieved the more easily. Chapter 2 deals with probability measures and includes a discussion of the fundamental concepts of probability theory. These concepts are formulated abstractly but without sacrificing intuition. The last chapter is devoted to infinite sums of independent real random variables. Each chapter is divided into sections that end with a set of problems with hints for solution.

This textbook will be particularly valuable to students of mathematics taking courses in probability theory who need a modern introduction to the subject that yet does not allow overemphasis on abstractness to cloud the issues involved.

Contents:

1 Finite trials 1

1.1 Probability spaces 1

1.2 Real random variables and random vectors 3

1.3 Mixing, direct composition, and tree composition 13

1.4 Conditional probabilities 24

1.5 Independence 27

1.6 Independent random variables 32

1.7 The law of large numbers 35

2 Probability measures 38

2.1 General trials and probability measures 38

2.2 The extension theorem of probability measures 46

2.3 Direct products of probability measures 53

2.4 Standard probability spaces 60

2.5 One-dimensional distributions 67

2.6 Characteristic functions 80

2.7 The weak topology in the distributions 97

2.8 D-Dimensional distributions 102

2.9 Infinite-dimensional distributions 105

3 Fundamental concepts in probability theory 110

3.1 Separable perfect probability measures 110

3.2 Events and random variables 114

3.3 Decompositions and o-algebras 123

3.4 Independence 129

3.5 Conditional probability measures 136

3.6 Properties of conditional probability measures 145

3.7 Real random variables 148

3.8 Conditional mean operators 157

4 Sums of independent random variables 165

4.1 General remarks 165

4.2 Convergent series of independent random variables 170

4.3 Central values and dispersions 175

4.4 Divergent series of independent random variables 184

4.5 Strong law of large numbers 186

4.6 Central limit theorems 191

4.7 The law of iterated logarithms 199

4.8 Gauss's theory of errors 206

4.9 Poisson's law of rare events 209.

Notes:

Translation of: Kakuritsuron. 1-4-shō.

Includes index.

ISBN:

0521264189

OCLC:

10123440