Abstract
In this note we look into detail into the box-counting dimension of subordinators. Given that X is a non-decreasing Levy process which is not Compound Poisson process we show that in the limit, a.s., the minimum number of boxes of size $a$ that cover the range of $(X_s)_{s\leq t}$ is a.s. of order $t/U(a)$, where U is the potential function of X. This is a more rened result than the lower and upper index of the box-counting dimension computed by Jean Bertoin in his 1999 book, which deals with the asymptotic of the number of boxes at logarithmic scale.
Citation
Mladen Savov. "On the range of subordinators." Electron. Commun. Probab. 19 1 - 10, 2014. https://doi.org/10.1214/ECP.v19-3629
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