R�gularit� de processus gaussiens

Abstract

Sommaire. On 6tudie la continuit6 presque sfire ou la majoration presque sfire des trajectoires de certains processus gaussiens en fonction de la r6gularit6 de leur covariance: on utilise la notion d'espace d'Orlicz pour obtenir des r6sultals plus fins que les r6sultats classiques. En appendice, on donne une d6monstration 616mentaire de l'int6grabilit6 des vecteurs aleatoires gaussiens fi valeurs dans les espaces vectoriels g6n6raux.

Sample functions of Pólya processes

“…The following example shows the sample unboundedness of Polya process, being defined by the / which does not satisfy conditions in the Theorem 1, that is, / is not continuous at the two points and not of bounded variation. This Polya process is an example of the class considered by Fernique [3].…”

On the upper and lower class for stationary Gaussian processes

“…We denote the correlation function of X by p and assume p(0)=l, and accordingly <r2(«) = 2(l -p(h)). (4) The superscript u expresses "with respect to the uniform continuity". From this theorem, we can easily see the following Corollary 1.1.…”

“…According to Lemma 15, it holds that under the assumption (4) 00 [(lOg P)l '«] (59) 2 2 P(E(p;k))<oe.…”

Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions

“…-If {e^} were normal random variables this would be a consequence of Fernique's theorem [4]. Dudley's theorem implies Fernique's theorem (see [2], Theorem 7.1 and [10]) and the proof of Dudley's theorem (as well as the proof of Fernique's theorem) depends only on the increments of X(() satisfying (3.2) rather than on the somewhat finer estimate available for Gaussian random variables.…”