This chapter provides a systematic introduction to the book.

This chapter introduces sub-Gaussian and sub-exponential distributions and develops basic concentration inequalities. We prove the Hoeffding, Chernoff, Bernstein, and Khintchine inequalities. Applications include robust mean estimation and analyzing degrees in random graphs. The exercises explore Mills ratio, small ball probabilities, Le Cam’s two-point method, the expander mixing lemma for random graphs, stochastic dominance, Orlicz norms, and the Bennett inequality.

This chapter demonstrates that finite convex risk measures on Lp spaces (for p ∈ [1, ∞) are inherently lower semicontinuous, ensuring the validity of their dual representations, while for L∞ spaces, the Fatou property is required.

Most of the material in this chapter is from basic analysis and probability courses. Key concepts and results are recalled here, including convexity, norms and inner products, random variables and random vectors, union bound, conditioning, basic inequalities (Jensen, Minkowski, Cauchy–Schwarz, Hölder, Markov, and Chebyshev), the integrated tail formula, the law of large numbers, the central limit theorem, normal and Poisson distributions, and handy bounds on the factorial.

This chapter presents some foundational methods for bounding random processes. We begin with the chaining technique and prove the Dudley inequality, which bounds a random process using covering numbers. Applications include Monte Carlo integration and uniform bounds for empirical processes. We then develop VC (Vapnik– Chervonenkis) theory, offering combinatorial insights into random processes and applying it to statistical learning. Building on chaining, we introduce generic chaining to obtain optimal two-sided bounds using Talagrand’s g2 functional. A key consequence is the Talagrand comparison inequality, a generalization of the Sudakov–Fernique inequality for sub-Gaussian processes. This is used to derive the Chevet inequality, a powerful tool for analyzing random bilinear forms over general sets. Exercises explore the Lipschitz law of large numbers in higher dimensions, one-bit quantization, and the small ball method for heavy-tailed random matrices.

This chapter begins with Maurey’s empirical method – a probabilistic approach to constructing economical convex combinations. We apply it to bound covering numbers and the volumes of polytopes, revealing their counterintuitive behavior in high dimensions. The exercises refine these bounds and culminate in the Carl–Pajor theorem on the volume of polytopes.

This chapter introduces several basic tools in high-dimensional probability: decoupling, concentration for quadratic forms (the Hanson–Wright inequality), symmetrization, and contraction. These techniques are illustrated through estimates of the operator norm of a random matrix. This is applied to matrix completion, where the goal is to recover a low-rank matrix from a random subset of its entries. Exercises explore variants of the Hanson–Wright inequality, mean estimation, concentration of the norm for anisotropic random vectors, distances to subspaces, graph cutting, the concept of type in normed spaces, non-Euclidean versions of the approximate Caratheodory theorem, and covariance estimation.

This chapter introduces financial risk as a random variable representing uncertain future gains or losses. A risk measure quantifies this uncertainty by mapping random variables to real numbers. The prominent example of Value-at-Risk (V@R) is discussed.

The Average-Value-at-Risk (AV@R) has emerged as a superior, coherent risk measure that accounts for the magnitude of potential losses beyond a given quantile and consistently favors diversification.

This chapter begins the study of random vectors in high dimensions, starting by showing their norm concentrates. We give a probabilistic proof of the Grothendieck inequality and apply it to semidefinite optimization. We explore a semidefinite relaxation for the maximum cut, presenting the Goemans–Williamson randomized approximation algorithm. We also give an alternative proof of the Grothendieck inequality with nearly the best known constant using the kernel trick, a method widely used in machine learning. The exercises explore invariant ensembles of random matrix theory, various versions of the Grothendieck inequality, semidefinite relaxations, and the notion of entropy.