such as: Borel-Cantelli lemma, monotone and dominated convergence
theorems, Borel measure, conditional expectation, etc. I also try to
firm up the students’ grasp of the advanced calculus throughout the
book. For example, analysis of simple random walk leads to Stirling’s
formula whose proof uses Taylor’s theorem with remainder.
It is hoped that this book will be interesting to undergraduates,
especially those considering graduate studies, as well as to graduate
students and faculty whose specialty is not probability or analysis.
This book could be used for advanced seminars or for independent
reading. There are a number of exercises at the end of each section.
They vary in diﬃculty and some of them are at the challenging level
that corresponds to summer projects for undergraduates at the REU.
I would like to thank Marcelo Alvisio, Laurence Field, and Jacob
Perlman for their comments on a draft of this book and the National
Science Foundation for continued support.