This chapter presents the matrix deviation inequality, a uniform deviation bound for random matrices over general sets. Applications include two-sided bounds for random matrices, refined estimates for random projections, covariance estimation in low dimensions, and an extension of the Johnson–Lindenstrauss lemma to infinite sets. We prove two geometric results: the M* bound, which shows how random slicing shrinks high-dimensional sets, and the escape theorem, which shows how slicing can completely miss them. These tools are applied to a fundamental data science task – learning structured high-dimensional linear models. We extend the matrix deviation inequality to arbitrary norms and use it to strengthen the Chevet inequality and derive the Dvoretzky– Milman theorem, which states that random low-dimensional projections of high-dimensional sets appear nearly round. Exercises cover matrix and process-level deviation bounds, high-dimensional estimation techniques such as the Lasso for sparse regression, the Garnaev–Gluskin theorem on random slicing of the cross-polytope, and general-norm extensions of the Johnson–Lindenstrauss lemma.

On atomless probability spaces, all law-determined convex risk measures on Lp spaces can be represented as a supremum of integrals of Average-Value-at-Risk (AV@R) measures, demonstrating AV@R’s role as a fundamental building block.

Chapter 7 covers models with categorical endogenous variables. It examines the consequences of treating such variables as continuous and how to modify SEMs to take account of categorical variables. It begins with single equation regression-like models for binary, ordinal, and count variables and builds to multiequation models. It includes a polychoric correlation approach, models with exogenous observed variables, the treatment of missing values, and alternative modeling approaches for categorical variables.

This chapter introduces structural equation models (SEMs). It defines SEMs and outlines their history. It also presents several widespread misunderstandings about SEMs and presents their strengths and weaknesses. Finally, the chapter provides an outline of the remaining book chapters.

This chapter explores various constructions of risk measures, including spectral risk measures, distortion risk measures, and moment-based risk measures, as well as risk measures generated by expected losses.

This chapter introduces techniques for bounding random processes. We develop Gaussian interpolation to derive powerful comparison inequalities for Gaussian processes, including the Slepian, Sudakov–Fernique, and Gordon inequalities. We use this to get sharp bounds on the operator norm of Gaussian random matrices. We also prove the Sudakov lower bound using covering numbers. We introduce the concept of Gaussian width, which connects probabilistic and geometric perspectives, and apply it to analyze the size of random projections of high-dimensional sets. Exercises cover symmetrization and contraction inequalities for random processes, the Gordon min–max inequality, sharp bounds for Gaussian matrices, the nuclear norm, effective dimension, random projections, and matrix sketching.

This chapter demonstrates that coherent and comonotonic additive risk measures are characterized by Choquet integrals with respect to two-alternating (submodular or concave) non-additive measures.

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.