Supremum of a Process in Terms of Trees

In this paper we study the quantity E supt∈T Xt, where Xt is some random process. In the case of the Gaussian process, there is a natural sub-metric d defined on T . We find an upper bound in terms of labelled-covering trees of (T, d) and a lower bound in terms of packing trees (this uses the knowledge of packing numbers of subsets of T ). The two quantities are proved to be equivalent via a general result concerning packing trees and labelled-covering trees of a metric space. Instead of using the majorizing measure theory, all the results involve the language of entropy numbers. Part of the results can be extended to some more general processes which satisfy some concentration inequality.