Probability Theory, An Analytic View

A rigorous, yet entertaining, account of the analytic foundations on which Kolmogorov built the theory of probability.

Daniel W. Stroock (Author)

9781009549004, Cambridge University Press

Paperback / softback, published 21 November 2024

466 pages
25.4 x 17.7 x 2.5 cm, 0.85 kg

'Highly recommended.' M. Bona, Choice

The third edition of this highly regarded text provides a rigorous, yet entertaining, introduction to probability theory and the analytic ideas and tools on which the modern theory relies. The main changes are the inclusion of the Gaussian isoperimetric inequality plus many improvements and clarifications throughout the text. With more than 750 exercises, it is ideal for first-year graduate students with a good grasp of undergraduate probability theory and analysis. Starting with results about independent random variables, the author introduces weak convergence of measures and its application to the central limit theorem, and infinitely divisible laws and their associated stochastic processes. Conditional expectation and martingales follow before the context shifts to infinite dimensions, where Gaussian measures and weak convergence of measures are studied. The remainder is devoted to the mutually beneficial connection between probability theory and partial differential equations, culminating in an explanation of the relationship of Brownian motion to classical potential theory.

Notation
1. Sums of independent random variables
2. The central limit theorem
3. Infinitely divisible laws
4. Lévy processes
5. Conditioning and martingales
6. Some extensions and applications of martingale theory
7. Continuous parameter martingales
8. Gaussian measures on a Banach space
9. Convergence of measures on a Polish space
10. Wiener measure and partial differential equations
11. Some classical potential theory
References
Index.

Subject Areas: Calculus & mathematical analysis [PBK]