The Supremum of Some Canonical Processes

Abstract

1. Introduction. Consider a random variable (r.v.) h such that Eh2 l distributed like h. To each t = (tk)k>1 of ?2 one can associate a r.v. XI = Ek>l tkhk. This defines a stochastic process (Xt)t,e2. (By stochastic process we simply mean a collection of r.v.) This process could be called the canonical process associated to h. A long range (and rather ambitious) program would be to understand, given h, for which subsets T of ?2 the process (Xt)tCT is bounded. Let us first discuss the case where h is (centered) Gaussian. The special importance of that case stems from the fact that, for the question of boundedness discussed here, Gaussian processes of the type (Xt)tCT are the most general Gaussian processes. (This fact is closely related to the rotational invariance of Gaussian measures.) For a stochastic process (Xt)tcT, where the set T is possibly uncountable, we set