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On the range of subordinators. (English) Zbl 1329.60127
Summary: In this note, we look in detail into the box-counting dimension of subordinators. Given that \(X\) is a non-decreasing Lévy process which is not a 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 refined result than the lower and upper index of the box-counting dimension computed by J. Bertoin [Lect. Notes Math. 1717, 1–91 (1999; Zbl 0955.60046), Theorem 5.1], which deals with the asymptotics of the number of boxes at logarithmic scale.

MSC:
60G51 Processes with independent increments; Lévy processes
60F15 Strong limit theorems
52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
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