A Functional LIL for d-Dimensional Stable Processes; Invariance for Lévy- and Other Weakly Convergent Processes

Abstract

We establish a functional LIL for the maximal process M(t) :=sup 0≤s≤t ‖X(s)‖ of an ℝ d -valued α-stable Levy process X, provided X(1) has density bounded away from zero over some neighborhood of the origin. We also provide a broad invariance result governing a class independent-increment processes related to the domain of attraction of X(1). This breadth is particularly notable for two types of processes captured: First, it not only describes any partial sum process built from iid summands in the domain of normal attraction of X(1), but also addresses those with arbitrary iid summands in the full domain of attraction (here we give a technical condition necessary and sufficient for the partial sum process to share the exact LIL we prove for X). Second, it reveals that any Levy process L such that L(1) satisfies the technical condition just mentioned will also share the LIL of X.