Maximal Moments and Uniform Modulus of Continuity for Stable Random Fields

Abstract

In this work, we solve an open problem mentioned in Xiao (2010) and provide sharp bounds on the rate of growth of maximal moments for stationary symmetric stable random fields using structure theorem of finitely generated abelian groups and ergodic theory of quasi-invariant group actions. We also investigate the relationship between this rate of growth and the path regularity properties of self-similar stable random fields with stationary increments, and establish uniform modulus of continuity of such fields. In the process, a new notion of weak effective dimension is introduced for stable random fields and is connected to maximal moments and path properties. Our results establish a boundary between shorter and longer memory in relation to Holder continuity of S alpha S random fields confirming a conjecture of Samorodnitsky (2004).

Major revision: (1) The notion of weak effective dimension (not necessarily an integer) has been introduced extending the effective dimension (always an integer) of stable random fields. (2) Weak effective dimension has been used to give sharper results. (3) Asymptotics of maximal moments in the continuous parameter case has been added

Cited by in corpus (2)

• Group measure space construction, ergodicity and $W^{\ast }$-rigidity for stable random fields
• A large sample test for the length of memory of stationary symmetric stable random fields via nonsingular ${\mathbb{Z}}^d$-actions