Knill Oliver. Probability and Stochastic Processes with Applications

Unpublished, 2008. — 382 p.These notes grew from an introduction to probability theory taught during the first and second term of 1994 at Caltech. There was a mixed audience of undergraduates and graduate students in the first half of the course which covered Chapters 2 and 3, and mostly graduate students in the second part which covered Chapter 4 and two sections of Chapter 5.Having been online for many years on my personal web sites, the text got reviewed, corrected and indexed in the summer of 2006. It obtained some enhancements which benefited from some other teaching notes and research,
I wrote while teaching probability theory at the University of Arizona in Tucson or when incorporating probability in calculus courses at Caltech and Harvard University.Most of Chapter 2 is standard material and subject of virtually any course on probability theory. Also Chapters 3 and 4 is well covered by the literture but not in this combination.The last chapter “selected topics” got considerably extended in the summer of 2006. While in the original course, only localization and percolation problems were included, I added other topics like estimation theory, Vlasov dynamics, multi-dimensional moment problems, random maps, circle-valued random variables, the geometry of numbers, Diophantine equations and harmonic analysis. Some of this material is related to research I got interested in over time.While the text assumes no prerequisites in probability, a basic exposure to calculus and linear algebra is necessary. Some real analysis as well as some background in topology and functional analysis can be helpful.