From a stochastic maximal inequality to infinite-dimensional martingales, towards high-dimensional statistics

Abstract

A novel approach is proposed to establish a sharp upper bound on the expected supremum of a separable martingale random field, serving as an alternative to classical universal chaining-based methods. The proposed approach begins by deriving a new "stochastic maximal inequality" for a finite class of discrete-time martingales. This is achieved by using some variations of log-sum-exp and softmax functions, as well as martingale transforms, avoiding the simple use of the triangle inequalty. We apply this inequality to obtain a generalization of Lenglart's inequality for discrete-time martingales, extending it from the one-dimensional case to finite-dimensional settings, and further to certain infinite-dimensional cases through a "finite approximation device." The primary applications include several weak convergence theorems for sequences of separable martingale random fields under the uniform topology. In particular, new results are established for i.i.d. sequences, including a necessary and sufficient condition for classes of functions to possess the Donsker property. Additionally, we provide new moment bounds for the supremum of empirical processes indexed by classes of sets or functions. The results and methods presented in this paper are also expected to be highly useful for high-dimensional statistics, including LASSO and Dantzig selectors, as it is illustrated in the last part of this paper.