Probability Theory

A Comprehensive Course

  • Achim Klenke

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Achim Klenke
    Pages 1-45
  3. Achim Klenke
    Pages 47-75
  4. Achim Klenke
    Pages 77-84
  5. Achim Klenke
    Pages 85-99
  6. Achim Klenke
    Pages 101-130
  7. Achim Klenke
    Pages 131-143
  8. Achim Klenke
    Pages 145-168
  9. Achim Klenke
    Pages 169-188
  10. Achim Klenke
    Pages 189-203
  11. Achim Klenke
    Pages 205-215
  12. Achim Klenke
    Pages 231-243
  13. Achim Klenke
    Pages 245-271
  14. Achim Klenke
    Pages 273-293
  15. Achim Klenke
    Pages 331-349
  16. Achim Klenke
    Pages 351-388
  17. Achim Klenke
    Pages 389-410
  18. Achim Klenke
    Pages 411-438
  19. Achim Klenke
    Pages 439-456
  20. Achim Klenke
    Pages 457-508
  21. Achim Klenke
    Pages 509-519
  22. Achim Klenke
    Pages 521-541
  23. Achim Klenke
    Pages 543-561
  24. Achim Klenke
    Pages 563-588
  25. Achim Klenke
    Pages 589-611
  26. Back Matter
    Pages 613-638

About this book


This second edition of the popular textbook contains a comprehensive course in modern probability theory. Overall, probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us in understanding magnetism, amorphous media, genetic diversity and the perils of random developments at financial markets, and they guide us in constructing more efficient algorithms.
To address these concepts, the title covers a wide variety of topics, many of which are not usually found in introductory textbooks, such as:
• 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.


Brownian Motion Integration Theory Limit Theorems Markov Chains Martingales Measure Theory Percolation Poisson Point Process Statistical Physics Stochastic Differential Equations Stochastic Integral Stochastic Processes

Authors and affiliations

  • Achim Klenke
    • 1
  1. 1.Johannes Gutenberg-Universität Mainz Institut für MathematikMainzGermany

Bibliographic information

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