Geoffrey Grimmett
After a brief review of foundational aspects of probability theory,
the course will cover central topics of the modern theory together
with many applications. Prior experience of intermediate-level
probability would be useful but not indispensable (such as experience
of some or all of the material of the Part IIB course on `Probability
and Measure'). The course leads on to other Part 3 courses, especially
`Stochastic Calculus and Applications'.
It is anticipated that the following principal topics will form the
main elements of the course.
- Foundations of probability. Borel-Cantelli lemmas, zero-one
laws, integration and expectation, conditional expectation.
- Convergence of random variables. Modes of convergence,
strong law (Etemadi), central limit theorems, applications (perhaps
include Erdos-Kac CLT for number of prime divisors of a random
integer). Construction of Brownian Motion, and the Markov property.
- Discrete-time martingales. Submartingales and
supermartingales, convergence theorems, optional stopping, backward
martingales, concentration inequalities, applications to gambling,
operational research.
- Large deviations. Cramér's theorem, large deviation
principle, applications.
- Ergodic theory. Stationary sequences, ergodic theorems,
applications to random walks and analysis.
References
- 1.
- G. Grimmett and D. Stirzaker, Probability and Random
Processes, OUP, 1991.
- 2.
- D. Williams, Probability and Martingales, CUP,
1991.
- 3.
- P. Billingsley, Probability and Measure, Wiley,
1979.
- 4.
- S. Karlin and H. Taylor, A First Course in
Stochastic Processes, Academic Press, 1975.