Tail Behavior of Hölder Norms of Stochastic Processes and Weak Convergence of Maxima in Hölder Spaces

Abstract

Our main goal was to establish functional limit theorems for component–wise maxima of iid processes taking values in Hölder spaces. Given that the finite–dimensional distributions converge, the key technical challenge is to establish tightness. The classical tightness conditions of Lamperti apply, provided that one can control the tail–behavior of Hölder norms. We do so, by using a powerful isomorphism theorem due to Ciesielski, which relates Hölder norms to superma of sequences. As a consequence, we obtain estimates for the tail probabilities of Hölder norms for light, moderate and heavy–tailed stochastic processes. The established inequalities are of independent interest since they provide explicit bounds on the tail probabilities of Hölder norms and suprema of stochastic processes under simple conditions on the bivariate distributions. We illustrate the results with sufficient conditions for the Hölder regularity of several classes of doubly stochastic max–stable processes of Schlather and Brown–Resnick types. Some extensions to the case of weak convergence in Besov spaces are also considered. 2010 Mathematics Subject Classification: 60F17, 60G17, 60G70.