Foundations of Probability

by Renyi, Alfred

• ISBN13: 9780486462615
• ISBN10: 0486462617
• Format: Paperback
• Copyright: 2007-12-17
• Publisher: Dover Publications

Summary

Introducing many innovations in content and methods, this book involves the foundations, basic concepts, and fundamental results of probability theory. Geared toward readers seeking a firm basis for study of mathematical statistics or information theory, it also covers the mathematical notions of experiments and independence. 1970 edition.

Author Biography

Alfred Renyi was Director of the Mathematical Institute of the Hungarian Academy of Sciences.

Alfred Renyi: The Happy Mathematician
Alfred Renyi (1921–1970) was one of the giants of twentieth-century mathematics who, during his relatively short life, made major contributions to combinatorics, graph theory, number theory, and other fields.

Reviewing Probability Theory and Foundations of Probability simultaneously for the Bulletin of the American Mathematical Society in 1973, Alberto R. Galmarino wrote:

"Both books complement each other well and have, as said before, little overlap. They represent nearly opposite approaches to the question of how the theory should be presented to beginners. Rényi excels in both approaches. Probability Theory is an imposing textbook. Foundations is a masterpiece."

In the Author's Own Words:
"If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy."

"Can the difficulty of an exam be measured by how many bits of information a student would need to pass it? This may not be so absurd in the encyclopedic subjects but in mathematics it doesn't make any sense since things follow from each other and, in principle, whoever knows the bases knows everything. All of the results of a mathematical theorem are in the axioms of mathematics in embryonic form, aren't they?" — Alfred Rényi

Table of Contents

Experiments
The definition of an experiment	p. 1
Algebras of events as Boolean algebras	p. 6
Operations with experiments	p. 9
Canonical representation of polynomials of events	p. 12
Qualitative independence	p. 16
On the structure of algebras of events of a finite or denumerable basic space	p. 17
Random mappings and random variables	p. 20
Qualitative entropy and information	p. 23
Probability
The intuitive notion of probability	p. 33
Conditional probability spaces	p. 38
Probability spaces	p. 47
Some remarks on the history of probability theory	p. 53
Limits of conditional probability spaces	p. 57
Linear inequalities and identities of probability theory	p. 63
Random variables on probability spaces	p. 68
Random variables on conditional probability spaces	p. 72
Expectations and other characteristics of probability distributions	p. 74
Conditional expectations and other characteristics of random variables	p. 81
Inequalities concerning random variables	p. 85
Some remarks on the notion of random variables	p. 86
Independence
Independence of two events	p. 100
Independence of sequences of events	p. 104
Construction of a probability measure with respect to which qualitatively independent events are independent	p. 109
Product spaces	p. 119
Independent random variables	p. 122
Independence and orthogonality	p. 127
Independence and ergodic theory	p. 138
Independence and information	p. 146
Sufficient functions	p. 157
Markov chains	p. 159
The Laws of Chance
The nature of laws of chance	p. 174
Types of convergence of sequences of random variables	p. 175
Convergence of probability distributions	p. 180
The laws of large numbers	p. 195
Approximations to the binomial and multinomial distributions	p. 204
The Poisson process	p. 213
The central limit theorem	p. 223
Laws of fluctuation	p. 229
Dependence
Conditional expectations with respect to a [sigma]-algebra	p. 259
Martingales	p. 268
Inequalities for martingales	p. 273
A martingale-convergence theorem and its applications	p. 276
Existence theorems	p. 289
Limit theorems for Markov chains	p. 293
Stable sequences of events	p. 301
Mixing sequences of events	p. 308
Exchangeable events	p. 315
The invariance of limit theorems under change of measure	p. 322
Appendix A	p. 345
Appendix B	p. 354
Index	p. 360
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