Probability Theory

We will formulate and prove some of the major results of classical probability. These will include the ``laws of large numbers" (both weak and strong), the ``central limit theorem", and the ``law of the iterated logarithm". All of these results originated in the mathematics of gambling, in efforts to predict the frequency of (say) heads in repeated tosses of a coin. They answer questions like how many heads there will be 'on average', and how far from the 'average' values the actual frequencies are likely to be. In answering these questions, we will develop useful tools such as characteristic functions, and will consider various modes of convergence for random variables. We will study important classes of distributions, such as stable and infinitely divisible laws. We may also study martingales.

At a more advanced level, the mathematical foundation of probability is a subject called measure theory. This will not be required for our course. Instead, we will merely take from measure theory the technical results we shall need, and proceed to their interesting probabilistic consequences. Further study of measure theory is possible at the graduate level.

The prerequisite is an introductory course in probability. A course in real analysis would be helpful, but is not required. MATH4030.03 is crosslisted with MATH6600.03. Starting in 1996-1997, MATH3030.03 will be the required probability prerequisite.

The text will be "Probability and Random Processes", by Grimmett and Stirzaker.