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PRODUCT FRACTAL SETS DETERMINED BY STABLE PROCESSES

Part of: Classical measure theory Stochastic processes

Published online by Cambridge University Press:  26 February 2009

YAN-YAN HOU  and
MIN-ZHI ZHAO
YAN-YAN HOU*
Affiliation:
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, P. R. China (email: hyymath@hotmail.com)
MIN-ZHI ZHAO
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China (email: zhaomz@zju.edu.cn)
*
✉For correspondence; e-mail: hyymath@hotmail.com
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Abstract

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Let Xi be transient βi-stable processes on ℝdi, i=1,2. Assume further that X1 and X2 are independent. We shall find the exact Hausdorff measure function for the product sets R1(1)×R2(1), where . The result of Hu generalizes [Some fractal sets determined by stable processes, Probab. Theory Related Fields100 (1994), 205–225].

Keywords

stable processesexact Hausdorff measurerange

MSC classification

Secondary: 60G18: Self-similar processes 28A78: Hausdorff and packing measures
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 2 , April 2009 , pp. 201 - 212
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

Research supported by NSFC Grant 10601047 and 2008XNM06.

References

[1] Falconer, K. J., Fractal Geometry: Mathematical Foundations and Applications (Wiley, Chichester, 1990).Google Scholar
[2] Hu, X. Y., ‘Some fractal sets determined by stable processes’, Probab. Theory Related Fields 100 (1994), 205–225.CrossRefGoogle Scholar
[3] Pruitt, W. E. and Taylor, S. J., ‘Sample path properties of processes with stable components’, Z. Wahrsch. Verw. Gebiete 12 (1969), 267–289.CrossRefGoogle Scholar
[4] Rogers, C. A. and Taylor, S. J., ‘Functions continuous and singular with respect to a Hausdorff measure’, Mathematika 8 (1961), 1–31.CrossRefGoogle Scholar
[5] Taylor, S. J., ‘Sample path properties of a transient stable process’, J. Math. Mech. 16 (1967), 1229–1246.Google Scholar