First Look at Rigorous Probability Theory, a (2nd Edition)

Second Edition

By: Jeffrey S. Rosenthal

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Paperback | 15 November 2006 | Edition Number 2

At a Glance

Paperback
236 Pages

Edition Type
Revised

Dimensions(cm)
23 x 15.5 x 2

Paperback

RRP $59.99

$49.80

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This textbook is an introduction to probability theory using measure theory. It is designed for graduate students in a variety of fields (mathematics, statistics, economics, management, finance, computer science, and engineering) who require a working knowledge of probability theory that is mathematically precise, but without excessive technicalities. The text provides complete proofs of all the essential introductory results. Nevertheless, the treatment is focused and accessible, with the measure theory and mathematical details presented in terms of intuitive probabilistic concepts, rather than as separate, imposing subjects. In this new edition, many exercises and small additional topics have been added and existing ones expanded. The text strikes an appropriate balance, rigorously developing probability theory while avoiding unnecessary detail.

Industry Reviews

"This book is certainly well placed to establish itself as a core reading in measure-theoretic probability ... Yit is? delightful reading and a worthwhile addition to the existing literature."

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Preface to the First Edition	p. vii
Preface to the Second Edition	p. xi
The need for measure theory	p. 1
Various kinds of random variables	p. 1
The uniform distribution and non-measurable sets	p. 2
Exercises	p. 4
Section summary	p. 5
Probability triples	p. 7
Basic definition	p. 7
Constructing probability triples	p. 8
The Extension Theorem	p. 10
Constructing the Uniform[0, 1] distribution	p. 15
Extensions of the Extension Theorem	p. 18
Coin tossing and other measures	p. 20
Exercises	p. 23
Section summary	p. 27
Further probabilistic foundations	p. 29
Random variables	p. 29
Independence	p. 31
Continuity of probabilities	p. 33
Limit events	p. 34
Tail fields	p. 36
Exercises	p. 38
Section summary	p. 41
Expected values	p. 43
Simple random variables	p. 43
General non-negative random variables	p. 45
Arbitrary random variables	p. 49
The integration connection	p. 50
Exercises	p. 52
Section summary	p. 55
Inequalities and convergence	p. 57
Various inequalities	p. 57
Convergence of random variables	p. 58
Laws of large numbers	p. 60
Eliminating the moment conditions	p. 61
Exercises	p. 65
Section summary	p. 66
Distributions of random variables	p. 67
Change of variable theorem	p. 67
Examples of distributions	p. 69
Exercises	p. 71
Section summary	p. 72
Stochastic processes and gambling games	p. 73
A first existence theorem	p. 73
Gambling and gambler's ruin	p. 75
Gambling policies	p. 77
Exercises	p. 80
Section summary	p. 81
Discrete Markov chains	p. 83
A Markov chain existence theorem	p. 85
Transience, recurrence, and irreducibility	p. 86
Stationary distributions and convergence	p. 89
Existence of stationary distributions	p. 94
Exercises	p. 98
Section summary	p. 101
More probability theorems	p. 103
Limit theorems	p. 103
Differentiation of expectation	p. 106
Moment generating functions and large deviations	p. 107
Fubini's Theorem and convolution	p. 110
Exercises	p. 113
Section summary	p. 115
Weak convergence	p. 117
Equivalences of weak convergence	p. 117
Connections to other convergence	p. 119
Exercises	p. 121
Section summary	p. 122
Characteristic functions	p. 125
The continuity theorem	p. 126
The Central Limit Theorem	p. 133
Generalisations of the Central Limit Theorem	p. 135
Method of moments	p. 137
Exercises	p. 139
Section summary	p. 142
Decomposition of probability laws	p. 143
Lebesgue and Hahn decompositions	p. 143
Decomposition with general measures	p. 147
Exercises	p. 148
Section summary	p. 149
Conditional probability and expectation	p. 151
Conditioning on a random variable	p. 151
Conditioning on a sub-[sigma]-algebra	p. 155
Conditional variance	p. 157
Exercises	p. 158
Section summary	p. 160
Martingales	p. 161
Stopping times	p. 162
Martingale convergence	p. 168
Maximal inequality	p. 171
Exercises	p. 173
Section summary	p. 176
General stochastic processes	p. 177
Kolmogorov Existence Theorem	p. 177
Markov chains on general state spaces	p. 179
Continuous-time Markov processes	p. 182
Brownian motion as a limit	p. 186
Existence of Brownian motion	p. 188
Diffusions and stochastic integrals	p. 190
Ito's Lemma	p. 193
The Black-Scholes equation	p. 194
Section summary	p. 197
Mathematical Background	p. 199
Sets and functions	p. 199
Countable sets	p. 200
Epsilons and Limits	p. 202
Infimums and supremums	p. 204
Equivalence relations	p. 207
Bibliography	p. 209
Background in real analysis	p. 209
Undergraduate-level probability	p. 209
Graduate-level probability	p. 210
Pure measure theory	p. 210
Stochastic processes	p. 210
Mathematical finance	p. 211
Index	p. 213
Table of Contents provided by Ingram. All Rights Reserved.

First Look at Rigorous Probability Theory, a (2nd Edition)

Second Edition

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

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