The majorizing measure approach to the sample boundedness

TL;DR

This paper introduces an alternative approach to understanding the sample boundedness and continuity of stochastic processes, emphasizing the distribution of the argument maximum and establishing key bounds for Gaussian processes.

Contribution

It presents a new perspective on path regularity using the distribution of the maximum argument and proves the equivalence of majorizing measure functionals.

Findings

Provided a short proof of the exact lower bound on the expected supremum of Gaussian processes.

Established the equivalence between the usual majorizing measure functional and its conjugate.

Offered insights into the regularity of paths via the distribution of the maximum argument.

Abstract

In this paper we describe the alternative approach to the sample boundedness and continuity of stochastic processes. We show that the regularity of paths can be understood in terms of a distribution of the argument maximum. For a centered Gaussian process X(t), t\in T we obtain a short proof of the exact lower bound on E\sup_{t\in T} X(t). Finally we prove the equivalence of a usual majorizing measure functional to its conjugate version.