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Packing and covering indices for a general Lévy process. (English) Zbl 0862.60063

Summary: There has been substantial interest in the indices \(0\leq\beta''\leq\beta'\leq\beta\leq 2\), defined by R. M. Blumenthal and R. K. Getoor [J. Math. Mech. 10, 493-516 (1961; Zbl 0097.33703)], determined by a general Lévy process in \(\mathbb{R}^d\). The first author [ibid. 19, 371-378 (1969; Zbl 0192.54101)] defined an index \(\gamma\) which determines the covering dimension, and the second author [Math. Proc. Camb. Philos. Soc. 100, 383-406 (1986; Zbl 0622.60021)] showed that an index \(\gamma'\), first considered by W. J. Hendricks [Ann. Probab. 11, 589-592 (1983; Zbl 0532.60064)], determines the packing dimension for the trajectory. We prove that \(\beta/2\leq\gamma'\leq\min(\beta,d)\), and give examples to show that the whole range is attainable. However, we cannot completely determine the set of values of \((\gamma,\gamma',\beta)\) which can be attained as indices of some Lévy process.

MSC:

60J99 Markov processes
28A75 Length, area, volume, other geometric measure theory
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[15] MINNEAPOLIS, MINNESOTA 55455 UNIVERSITY OF VIRGINIA CHARLOTTESVILLE, VIRGINIA 22903 E-MAIL: sjt@virginia.edu
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