High Dimensional Probability

Abstract

About forty years ago it was realized by several researchers that the essential features of certain objects of Probability theory, notably Gaussian processes and limit theorems, may be better understood if they are considered in settings that do not impose structures extraneous to the problems at hand. For instance, in the case of sample continuity and boundedness of Gaussian processes, the essential feature is the metric or pseudometric structure induced on the index set by the covariance structure of the process, regardless of what the index set may be. This point of view ultimately led to the Fernique-Talagrand majorizing measure characterization of sample boundedness and continuity of Gaussian processes, thus solving an important problem posed by Kolmogorov. Similarly, separable Banach spaces provided a minimal setting for the law of large numbers, the central limit theorem and the law of the iterated logarithm, and this led to the elucidation of the minimal (necessary and/or sufficient) geometric properties of the space under which different forms of these theorems hold. However, in light of renewed interest in Empirical processes, a subject that has considerably influenced modern Statistics, one had to deal with a non-separable Banach space, namely $\mathcal{L}_{\infty}$. With separability discarded, the techniques developed for Gaussian processes and for limit theorems and inequalities in separable Banach spaces, together with combinatorial techniques, led to powerful inequalities and limit theorems for sums of independent bounded processes over general index sets, or, in other words, for general empirical processes.