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The packing measure of a general subordinator. (English) Zbl 0767.60009
Summary: Precise conditions are obtained for the packing measure of an arbitrary subordinator to be zero, positive and finite, or infinite. It develops that the packing measure problem for a subordinator $$X(t)$$ is equivalent to the upper local growth problem for $$Y(t)=\min(Y_ 1(t),Y_ 2(t))$$, where $$Y_ 1$$ and $$Y_ 2$$ are independent copies of $$X$$. A finite and positive packing measure is possible for subordinators “close to Cauchy”; for such a subordinator there is non-random concave upwards function that exactly describes the upper local growth of $$Y$$ (although, as is well-known, there is no such function for the subordinator $$X$$ itself).

##### MSC:
 60D05 Geometric probability and stochastic geometry
##### Keywords:
packing measure; subordinator; upper local growth
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##### References:
 [1] Fristedt, B.E., Pruitt, W.B.: Lower functions for increasing random walks and subordinators. Z. Wahrscheinlichkeitstheor. Verw. Geb.18, 167-182 (1971). · Zbl 0197.44204 · doi:10.1007/BF00563135 [2] Rezakhanlou, F., Taylor, S.J.: The packing measure of the graph of a stable process. Asterisque158, 341-362 (1988) · Zbl 0677.60082 [3] Taylor, S.J.: Sample path properties of a transient stable process. J. Math. Mech.16, 1229-1246 (1967) · Zbl 0178.19301 [4] Taylor, S.J., Tricot, C.: Packing measure and its evaluation for a Brownian path. Trans. Am. Math. Soc.288, 679-699 (1985) · Zbl 0537.28003 · doi:10.1090/S0002-9947-1985-0776398-8 [5] Taylor, S.J.: The measure theory of random fractals. Math. Proc. Camb. Philos. Soc.100, 383-406 (1986) · Zbl 0622.60021 · doi:10.1017/S0305004100066160 [6] Taylor, S.J.: The use of packing measure in the analysis of random sets. Proceedings of the 15th Symposium on Stochastic Processes and Applications. (Lect. Notes Math., pp. 214-222) Berlin Heidelberg New York: Springer 1987
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