This volume presents topics in probability theory covered during a
first-year graduate course given at the Courant Institute of Mathematical
Sciences. The necessary background material in measure theory is
developed, including the standard topics, such as extension theorem,
construction of measures, integration, product spaces, Radon–Nikodym
theorem, and conditional expectation.
In the first part of the book, characteristic functions are introduced,
followed by the study of weak convergence of probability distributions.
Then both the weak and strong limit theorems for sums of independent
random variables are proved, including the weak and strong laws of large
numbers, central limit theorems, laws of the iterated logarithm, and the
Kolmogorov three series theorem. The first part concludes with infinitely
divisible distributions and limit theorems for sums of uniformly
infinitesimal independent random variables.
The second part of the book mainly deals with dependent random variables,
particularly martingales and Markov chains. Topics include standard
results regarding discrete parameter martingales and Doob's inequalities.
The standard topics in Markov chains are treated, i.e., transience, and
null and positive recurrence. A varied collection of examples is given to
demonstrate the connection between martingales and Markov chains.
Additional topics covered in the book include stationary Gaussian
processes, ergodic theorems, dynamic programming, optimal stopping, and
filtering. A large number of examples and exercises is included. The book
is a suitable text for a first-year graduate course in probability.
Readership
Graduate students and research mathematicians interested in
probability theory and stochastic processes and in applications to economics,
and finance.