A First Look at Rigorous Probability Theory

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World Scientific, 2000 - 177 Seiten
This textbook is an introduction to rigorous probability theory using measure theory. It provides rigorous, complete proofs of all the essential introductory mathematical results of probability theory and measure theory. More advanced or specialized areas are entirely omitted or only hinted at. For example, the text includes a complete proof of the classical central limit theorem, including the necessary continuity theorem for characteristic functions, but the more general Lindeberg central limit theorem is only outlined and is not proved. Similarly, all necessary facts from measure theory are proved before they are used, but more abstract or advanced measure theory results are not included. Furthermore, measure theory is discussed as much as possible purely in terms of probability, as opposed to being treated as a separate subject which must be mastered before probability theory can be understood.
 

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Inhalt

The need for measure theory
1
Further probabilistic foundations
21
Expected values
32
Inequalities and laws of large numbers
43
Distributions of random variables
52
Stochastic processes and gambling games
58
Discrete Markov chains
68
Some further probability results
86
Characteristic functions
102
Decomposition of probability laws
118
Conditional probability and expectation
124
Martingales
131
Introduction to other stochastic processes
140
Mathematical Background
160
Bibliography
168
Urheberrecht

Weak convergence
96

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