Probability Theory: A Comprehensive Course

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Springer Science & Business Media, 30.08.2013 - 638 Seiten

This second edition of the popular textbook contains a comprehensive course in modern probability theory, covering a wide variety of topics which are not usually found in introductory textbooks, including:
• limit theorems for sums of random variables
• martingales
• percolation
• Markov chains and electrical networks
• construction of stochastic processes
• Poisson point process and infinite divisibility
• large deviation principles and statistical physics
• Brownian motion
• stochastic integral and stochastic differential equations.

The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in probability theory. This second edition has been carefully extended and includes many new features. It contains updated figures (over 50), computer simulations and some difficult proofs have been made more accessible. A wealth of examples and more than 270 exercises as well as biographic details of key mathematicians support and enliven the presentation. It will be of use to students and researchers in mathematics and statistics in physics, computer science, economics and biology.

 

Ausgewählte Seiten

Inhalt

Basic Measure Theory
1
Independence
47
Generating Functions
77
The Integral
85
Moments and Laws of Large Numbers
101
Convergence Theorems
131
LpSpaces and the RadonNikodym Theorem
145
Conditional Expectations
169
Markov Chains
351
Convergence of Markov Chains
389
Markov Chains and Electrical Networks
411
Ergodic Theory
439
Brownian Motion
457
Law of the Iterated Logarithm
509
Large Deviations
521
The Poisson Point Process
543

Martingales
189
Optional Sampling Theorems
205
Martingale Convergence Theorems and Their Applications
217
Backwards Martingales and Exchangeability
231
Convergence of Measures
245
Probability Measures on Product Spaces
273
Characteristic Functions and the Central Limit Theorem
294
Infinitely Divisible Distributions
331
The Itô Integral
563
Stochastic Differential Equations
589
Notation Index
613
References
617
Name Index
625
Subject Index
628
Urheberrecht

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Probability Theory: A Comprehensive Course
Achim Klenke
Keine Leseprobe verfügbar - 2013

Häufige Begriffe und Wortgruppen

algebra assume Borel bounded Brownian motion called characteristic function Clearly compute consider continuous local martingale convergence theorem convex Corollary countable define Definition denote density disjoint distribution function ergodic Example Exercise exists filtration finite measure fl-Co formula function f Hence holds implies inequality infinitely divisible large numbers law of large Lebesgue measure Lemma Let f Let Q Let X1 lim sup linear Markov chain Markov process Markov property martingale measurable map measurable space metric n-CO nonnegative o-algebra parameter Poisson Polish space probability measure probability space Probability Theory Proof Let random variables random walk real random variables recurrent Remark respect sequence solution square integrable square variation stochastic kernel stochastic process submartingale surely T-system transition matrix uniformly integrable unique values Xn)neN

Autoren-Profil (2013)

Achim Klenke is a professor at the Johannes Gutenberg University in Mainz, Germany.

Bibliografische Informationen

TitelProbability Theory: A Comprehensive Course
Autor/inAchim Klenke
Ausgabe2, illustriert
VerlagSpringer Science & Business Media, 2013
ISBN1447153618, 9781447153610
Länge638 Seiten
  
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