On the range of subordinators

Mladen Savov
2014 Electronic Communications in Probability  
In this note we look into detail at the box-counting dimension of subordinators. Given that X is a non-decreasing Levy process, which is not a compound Poisson process, we show that in the limit, the minimum number of boxes of size δ that cover the range of (X_s)_s≤ t is a.s. of order t/U(δ), where U is the potential function of X. This is a more refined result than the lower and upper index of the box-counting dimension computed in the literature which deals with the asymptotic number of boxes at logarithmic scale.
doi:10.1214/ecp.v19-3629 fatcat:q63pje4rkjbsxgupr3wjqkugfq