Ramon van Handel Probability in High Dimension

Probability in High Dimension Report Title Lecture notes from course ORF 570 Probability in High Dimension (educational material made freely available on my website) Ramon van Handel Probability in High Dimension ORF 570 Lecture Notes Princeton University This version: June 30, 2014 Preface These lecture notes were written for the course ORF 570: Probability in High Dimension that I taught at Princeton in the Spring 2014 semester. The aim was to introduce in as cohesive a manner as I could manage a set of methods, many of which have their origin in probability in Banach spaces, that arise across a broad range of contemporary problems in di↵erent areas. The notes are necessarily incomplete. The ambitious syllabus for the course was laughably beyond the scope of Princeton’s 12-week semester. As a result, there are regrettable omissions, as well as many fascinating topics that I would have liked to but could not cover in the context of this course. These include: a. Bernstein’s inequality does not appear anywhere in these notes (disgraceful!), nor do any Bernstein-type concentration inequalities (such as concentration of the exponential distribution and Talagrand’s concentration inequality for empirical processes) and the notion of modified log-Sobolev inequalities. These should be included at the end of Part I. b. Chaining with adaptive truncation and entropy with brackets. Beyond being a classical topic in empirical process theory, the power of the idea of adaptive truncation has again proven its value in the recent solution of the long-standing Bernoulli problem due to Bednorz and Lata la. c. Universality (prematurely included in Chapter 1 as a topic to be covered though I did not have time to do so) and an introduction to Stein’s method. d. Hypercontractivity and its applications, particularly to concentration inequalities and to sharp thresholds (the latter should be promoted to a fourth “general principle” in Chapter 1 in view of the ubiquity of phase transition phenomena in high-dimensional problems). e. No doubt this list will grow even longer if I don’t stop typing. Hopefully the opportunity will arise in the future to fill in some of these gaps, in which case I will post an updated version of these notes on my website. For now, as always, these notes are made available as-is. VIII Preface Please note that these are lecture notes, not a monograph. Many important ideas that I did not have the time to cover are included as problems at the end of each section. Doing the problems is the best way to learn the material. To avoid distraction I have on a few occasions ignored some minor technical issues (such as measurability issues of empirical processes or domain issues of Markov generators), but I have tried to give the reader a fair warning when this is the case. The notes at the end of each chapter do not claim to give a comprehensive historical account, but rather to indicate the immediate origin of the material that I used and to serve as a starting point for further reading. Many thanks are due to the 30 or so regular participants of the course. These lecture notes are loosely based on notes scribed by the students during the lectures. While they have been almost entirely rewritten, the scribe notes served as a crucial motivation to keep writing. I am particularly grateful to Maria Avdeeva, Mark Cerenzia, Jacob Funk, Danny Gitelman, Max Goer, Jiequn Han, Daniel Jiang, Mitchell Johnston, Haruko Kato, George Kerchev, Dan Lacker, Che-Yu Liu, Yuan Liu, Huanran Lu, Junwei Lu, Tengyu Ma, Efe Onaran, Zhaonan Qu, Patrick Rebeschini, Max Simchowitz, Weichen Wang, Igor Zabukovec, Tianqi Zhao, and Ziwei Zhu for serving as scribes. Princeton, June 2014