Regularity of Gaussian and Stable Processes

Abstract

In the preceding chapter, we presented some sufficient metric entropy and majorizing measure conditions for the sample boundedness and continuity of random processes satisfying incremental conditions. In particular, these results were applied to Gaussian random processes in Section 11.3. The main concern of this chapter is necessity. We will see indeed, as one of the main results, that the sufficient majorizing measure condition for a Gaussian process to be almost surely bounded or continuous is actually also necessary. This characterization thus provides a complete understanding of the regularity properties of Gaussian paths. The arguments of proof rely heavily on the basic ultrametric structure which lies behind a majorizing measure condition.