Probability Theory I

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Springer Science & Business Media, Mar 29, 1977 - Mathematics - 428 pages
This fourth edition contains several additions. The main ones con cern three closely related topics: Brownian motion, functional limit distributions, and random walks. Besides the power and ingenuity of their methods and the depth and beauty of their results, their importance is fast growing in Analysis as well as in theoretical and applied Proba bility. These additions increased the book to an unwieldy size and it had to be split into two volumes. About half of the first volume is devoted to an elementary introduc tion, then to mathematical foundations and basic probability concepts and tools. The second half is devoted to a detailed study of Independ ence which played and continues to playa central role both by itself and as a catalyst. The main additions consist of a section on convergence of probabilities on metric spaces and a chapter whose first section on domains of attrac tion completes the study of the Central limit problem, while the second one is devoted to random walks. About a third of the second volume is devoted to conditioning and properties of sequences of various types of dependence. The other two thirds are devoted to random functions; the last Part on Elements of random analysis is more sophisticated. The main addition consists of a chapter on Brownian motion and limit distributions.
 

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Contents

ELEMENTARY PROBABILITY THEORY
1
Axioms Independence and the Bernoulli Case
8
Dependence and Chains
24
Markov Processes
28
Complements and Details
42
NOTIONS OF MEASURE THEORY
53
SECtION PACE
59
2 Topological Spaces
65
Probability Laws and Types of Laws
214
SECtION PACE
226
INDEPENDENCE
233
Convergence and Stability of Sums Centering
243
Second Order Properties
247
18 Convergence and Stability of Sums Centering
255
19 Exponential Bounds and Normed Sums
266
central limit problem
280

Additive Set Functions
83
Complements and Details
100
Measure and Convergences
111
Integration
118
Indefinite Integrals Iterated Integrals
130
Complements and Details
139
SECtION IAC
149
Probability Distributions
168
Complements and Details
174
12 Convergence of Probabilities on Metric Spaces
189
Characteristic Functions and Distribution Functions
198
Evolution of the Problem
286
section pace
300
23 Solution of the Central Limit Problem
308
24 Normed Sums
331
Foundations Martingales and Decomposability
334
independent identically distributed summands
353
Random Walk
368
Brownian Motion and Limit Distributions
380
Bibliography
407
Index
413
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