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

Joshua Rushton ()
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Joshua Rushton: University of Wisconsin

Journal of Theoretical Probability, 2007, vol. 20, issue 3, 397-427

Abstract: 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 Lévy 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 Lévy process L such that L(1) satisfies the technical condition just mentioned will also share the LIL of X.

Keywords: α-stable Lévy process; Functional LIL; Small deviations; Invariance principle (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/s10959-007-0059-0

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