A discrete fractal in $\textbf {Z}^ 1_ +$
HTML articles powered by AMS MathViewer
- by Davar Khoshnevisan PDF
- Proc. Amer. Math. Soc. 120 (1994), 577-584 Request permission
Abstract:
In this paper, we show that the level sets of mean zero finite variance random walks in ${\mathbb {R}^1}$ form a discrete fractal in the sense of Barlow and Taylor. Analogously to the Brownian motion result, the Hausdorff dimension of the level sets is almost surely equal to $\tfrac {1} {2}$.References
- N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871, DOI 10.1017/CBO9780511721434
- M. T. Barlow and S. J. Taylor, Fractional dimension of sets in discrete spaces, J. Phys. A 22 (1989), no. 13, 2621–2628. With a reply by J. Naudts. MR 1003752, DOI 10.1088/0305-4470/22/13/053
- Martin T. Barlow and S. James Taylor, Defining fractal subsets of $\textbf {Z}^d$, Proc. London Math. Soc. (3) 64 (1992), no. 1, 125–152. MR 1132857, DOI 10.1112/plms/s3-64.1.125
- Claude Dellacherie and Paul-André Meyer, Probabilities and potential. B, North-Holland Mathematics Studies, vol. 72, North-Holland Publishing Co., Amsterdam, 1982. Theory of martingales; Translated from the French by J. P. Wilson. MR 745449
- Bert E. Fristedt and William E. Pruitt, Lower functions for increasing random walks and subordinators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 18 (1971), 167–182. MR 292163, DOI 10.1007/BF00563135
- Harry Kesten, An iterated logarithm law for local time, Duke Math. J. 32 (1965), 447–456. MR 178494
- Michael B. Marcus and Jay Rosen, Laws of the iterated logarithm for the local times of symmetric Levy processes and recurrent random walks, Ann. Probab. 22 (1994), no. 2, 626–658. MR 1288125
- J. Naudts, Dimension of discrete fractal spaces, J. Phys. A 21 (1988), no. 2, 447–452. MR 940591, DOI 10.1088/0305-4470/21/2/024
- Charles Stone, A local limit theorem for nonlattice multi-dimensional distribution functions, Ann. Math. Statist. 36 (1965), 546–551. MR 175166, DOI 10.1214/aoms/1177700165
- S. J. Taylor and J. G. Wendel, The exact Hausdorff measure of the zero set of a stable process, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 6 (1966), 170–180. MR 210196, DOI 10.1007/BF00537139
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 120 (1994), 577-584
- MSC: Primary 60J15; Secondary 28A80
- DOI: https://doi.org/10.1090/S0002-9939-1994-1185269-8
- MathSciNet review: 1185269