An Introduction to Probability Theory

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Paperback | 17 December 1984

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224 Pages

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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.

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Preface	p. v
Notation and abbreviations	p. vii
Finite trials	p. 1
Probability spaces	p. 1
Real random variables and random vectors	p. 3
Mixing, direct composition, and tree composition	p. 13
Conditional probabilities	p. 24
Independence	p. 27
Independent random variables	p. 32
The law of large numbers	p. 35
Probability measures	p. 38
General trials and probability measures	p. 38
The extension theorem of probability measures	p. 46
Direct products of probability measures	p. 53
Standard probability spaces	p. 60
One-dimensional distributions	p. 67
Characteristic functions	p. 80
The weak topology in the distributions	p. 97
D-Dimensional distributions	p. 102
Infinite-dimensional distributions	p. 105
Fundamental concepts in probability theory	p. 110
Separable perfect probability measures	p. 110
Events and random variables	p. 114
Decompositions and o-algebras	p. 123
Independence	p. 129
Conditional probability measures	p. 136
Properties of conditional probability measures	p. 145
Real random variables	p. 148
Conditional mean operators	p. 157
Sums of independent random variables	p. 165
General remarks	p. 165
Convergent series of independent random variables	p. 170
Central values and dispersions	p. 175
Divergent series of independent random variables	p. 184
Strong law of large numbers	p. 186
Central limit theorems	p. 191
The law of iterated logarithms	p. 199
Gauss's theory of errors	p. 206
Poisson's law of rare events	p. 209
Index	p. 212
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An Introduction to Probability Theory

At a Glance

Paperback

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