David Pollard's webpage

⑤  Probability tools, tricks, and miracles,
An unpublished book, aimed at new researchers who want to understand some of the methods that have been developed over the past half century. Originally it was intended as an update of my first two books, focusing mainly of empirical processes, but over the years it evolved into a more general discussion. Some topics:

• Calculus + convexity: a very useful and effective combination. Simple tricks to replace brute force. Chapter 2.
• Precise behavior of the N(0,1) tail probabilities, with an application to Stein's method for norml approximation. §3.4
• Tail bounds and relationship between Binomial, Poisson-Binomial, and Hypergeometric. A beatiful result of Hoeffding proved using a method invented by Chebyshev nearly 200 years ago. §§3.7, 3.8; Chapter 4.
• Quick and effective methods for controlling (maximal inequalities) stochastic processes by means of Orlicz norms. Chapter 5.
• For the multivariate normal: tail bounds, comparison inequalities, concentration. Discussion and comparison of three difference versions the path methods. Chapter 6.
• Subgaussian. Standard inequalities (Hoeffding for sums of independent variables or martingales) that every probabilist should know. Chapter 7.
• Bennett inequalities for independent summands or martingales, with an application to the Kim-Vu concentration results for random polynomials. §§8.3,8.4.
• The justification for the typical remark ``without loss of generality we may assume the index set is at worst countably infinite" when studying stochastic processes indexed by general metric spaces. Chapter 9.
• Chaining: Why does it work? Classical covering/packing approaches. How to choose useful sequences of constants. Oscillation bounds constructed as an exercise in approximation by finite sets of fixed size. Chapter 10.
• Methods of Fernique and Talagrand for construction of chaining frameworks. Relationship between majorizing measures, weighted chaining frameworks, and admissible sequences of partitions. Insights provided by particularly instructive variation on a classical example. Chapter 11.
• Weighted sequences of partitions for gaussian lower bounds. Constructions as algorithms. What is going on? Chapter 12.
• Symmetrization in the broad sense. How R. A. Fisher's ideas about experimental design 90 years ago anticipated the use of random ±1 variables (aka `Rademachers'). Maximal inqualities for sums of independent stochastic processes. Chapter 13.
• VC sets. What's going on? Downshifting on matrices filled with 0's and 1's as a way to prove the famous VC bound. Shatter dimension of subsets of {0,1}ⁿ and Haussler's interpretation via orientation of edges. (Very clever stuff.) Chapter 14.
• Surround dimension (aka fat shattering) for subsets of euclidean spaces and the consequences fo packing numbers. (Beautiful ideas about counting lattice points by Vershynin et al.) Chapter 15.