First Look at Rigorous Probability Theory

ISBN-10: 9812703713
ISBN-13: 9789812703712
Edition: 2nd 2006
List price: $33.00 Buy it from $18.87
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Description: 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  More...

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Book details

List price: $33.00
Edition: 2nd
Copyright year: 2006
Publisher: World Scientific Publishing Company, Incorporated
Binding: Paperback
Pages: 219
Size: 6.00" wide x 8.75" long x 0.75" tall
Weight: 0.946
Language: English

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.

Preface to the First Edition
Preface to the Second Edition
The need for measure theory
Various kinds of random variables
The uniform distribution and non-measurable sets
Exercises
Section summary
Probability triples
Basic definition
Constructing probability triples
The Extension Theorem
Constructing the Uniform[0, 1] distribution
Extensions of the Extension Theorem
Coin tossing and other measures
Exercises
Section summary
Further probabilistic foundations
Random variables
Independence
Continuity of probabilities
Limit events
Tail fields
Exercises
Section summary
Expected values
Simple random variables
General non-negative random variables
Arbitrary random variables
The integration connection
Exercises
Section summary
Inequalities and convergence
Various inequalities
Convergence of random variables
Laws of large numbers
Eliminating the moment conditions
Exercises
Section summary
Distributions of random variables
Change of variable theorem
Examples of distributions
Exercises
Section summary
Stochastic processes and gambling games
A first existence theorem
Gambling and gambler's ruin
Gambling policies
Exercises
Section summary
Discrete Markov chains
A Markov chain existence theorem
Transience, recurrence, and irreducibility
Stationary distributions and convergence
Existence of stationary distributions
Exercises
Section summary
More probability theorems
Limit theorems
Differentiation of expectation
Moment generating functions and large deviations
Fubini's Theorem and convolution
Exercises
Section summary
Weak convergence
Equivalences of weak convergence
Connections to other convergence
Exercises
Section summary
Characteristic functions
The continuity theorem
The Central Limit Theorem
Generalisations of the Central Limit Theorem
Method of moments
Exercises
Section summary
Decomposition of probability laws
Lebesgue and Hahn decompositions
Decomposition with general measures
Exercises
Section summary
Conditional probability and expectation
Conditioning on a random variable
Conditioning on a sub-[sigma]-algebra
Conditional variance
Exercises
Section summary
Martingales
Stopping times
Martingale convergence
Maximal inequality
Exercises
Section summary
General stochastic processes
Kolmogorov Existence Theorem
Markov chains on general state spaces
Continuous-time Markov processes
Brownian motion as a limit
Existence of Brownian motion
Diffusions and stochastic integrals
Ito's Lemma
The Black-Scholes equation
Section summary
Mathematical Background
Sets and functions
Countable sets
Epsilons and Limits
Infimums and supremums
Equivalence relations
Bibliography
Background in real analysis
Undergraduate-level probability
Graduate-level probability
Pure measure theory
Stochastic processes
Mathematical finance
Index

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