Abstract
Let X=(X(t)) t∈T be a symmetric α-stable, 0<α<2, process with paths in the dual E * of a certain Banach space E. Then there exists a (bounded, linear) operator u from E into some L α (S,σ) generating X in a canonical way. The aim of this paper is to compare the degree of compactness of u with the small deviation (ball) behavior of \(\phi (\varepsilon ) = - \log \mathbb{P}(\left\| X \right\|_{E^* } < \varepsilon )\) as ε→0. In particular, we prove that a lower bound for the metric entropy of u implies a lower bound for φ(ε) under an additional assumption on E. As applications we obtain upper small deviation estimates for weighted α-stable Levy motions, linear fractional α-stable motions and d-dimensional α-stable sheets. Our results rest upon an integral representation of L α -valued operators as well as on small deviation results for Gaussian processes due to Kuelbs and Li and to the authors.
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Li, W.V., Linde, W. Small Deviations of Stable Processes via Metric Entropy. Journal of Theoretical Probability 17, 261–284 (2004). https://doi.org/10.1023/B:JOTP.0000020484.26184.c4
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DOI: https://doi.org/10.1023/B:JOTP.0000020484.26184.c4