The packing indices for some Lévy processes

Abstract

Let X = { X t , t ≥ 0 } be a Lévy process, and γ 0 = sup { α ≥ 0 : lim inf a → 0 a − α E T ( a , 1 ) < ∞ } , where T ( a , 1 ) = ∫ 0 1 I { | X t | ≤ a } d t . Taylor (1986) showed that the packing dimension of the trajectory of X is γ 0 . An open question from Pruitt and Taylor (1996) is: Is it true that γ 0 = inf { α ≥ 0 : a − α T ( a , 1 ) → ∞ a.s. as a → 0 } ? In this paper, we give some answers to this question.