Random Processes

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.