## 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

### Keywords:

Lévy process; Hausdorff dimension; packing dimension

### Citations:

Zbl 0097.33703; Zbl 0192.54101; Zbl 0622.60021; Zbl 0532.60064
Full Text:

### References:

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