Summary
Let {X t } be a ℝ1 process with stationary independent increments and its Lévy measurev be given byv{y∶y>x}=x −αL 1 (x), v{y∶y<−x}=x −αL 2 (x) whereL 1,L 2 are slowly varying at 0 and ∞ and 0<α≦1. We construct two types of a nondecreasing functionh(t) depending on 0<α<1 or α=1 such that lim inf\(\mathop {\sup }\limits_{0 \leqq s \leqq t} |X_S |/h(t) = C\) a.s. ast→ 0 andt→∞ for some positive finite constantC.
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This research is partialy supported by a grant from Korea University
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Wee, IS. Lower functions for asymmetric Lévy processes. Probab. Th. Rel. Fields 85, 469–488 (1990). https://doi.org/10.1007/BF01203165
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DOI: https://doi.org/10.1007/BF01203165