Exact packing measure for the product set of the ranges of stable component processes

doi:10.7523/j.issn.2095-6134.2014.05.001

Abstract

Abstract:

Let {X¹(t₁),t₁≥0}, …,{X^p(t_p),t_p≥0} be p independent stable component processes in R^d. We obtain the law of the iterated logarithm for the measure on the product set ∏ _i=1^pXⁱ[0,1]. By applying the density theorem, we derive the exact packing measure of the product set. As a special case, we deduce the exact packing measure of the product set of ranges and graphs for p independent stable processes in R^d.

Key words: stable component process, the law of the iterated logarithm, packing measure

CLC Number:

O211.6

LIANG Longyue. Exact packing measure for the product set of the ranges of stable component processes[J]. , 2014, 31(5): 577-585.

References

[1] Taylor S J. Sample path properties of a transient stable process[J]. J Math Mech, 1967, 16: 1 229-1 246.

[2] Pruitt W E, Taylor S J. Sample path properties of processes with stable components[J]. Z Wahrsch Verw Gebiete, 1969, 12: 267-289.

[3] Port S C, Stone C J. Infinitely divisible processes and their potential theory Ⅰ, Ⅱ[J]. Ann Inst Fourier, 1971, 21(2): 157-275 and 1971, 21(4): 176-265.

[4] 赵兴球. 稳定分量过程的图集的 Packing 测度[J]. 数学年刊, 1995, 16A 3: 344-349.

[5] Taylor S J, Tricot C. Packing measure, and its evaluation for a Brownian path[J]. Trans Amer Math Soc, 1985, 288: 679-699.

[6] Taylor S J. The use of packing measure in the analysis of random sets[J]. Stochastic Processes and Their Applications, Lecture Notes in Math, vol Springer, 1985, 1203: 214-222.

[7] Jain N, Pruitt W E. The correct measure function for the graph of a transient stable process[J]. Z Wahrsch verw Geb, 1968, 9: 131-138.

[8] Taylor S J. The measure theory of random fractals[J]. Math Proc Camb Philos Soc, 1986, 100: 383-406.

[9] Xiao Y M. Random fractals and Markov processes[G]//Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot. Proc Symposia Pure Math, 2004, 72: 261-338.

[10] Besicovitch A S, Moran P A. The measure of product and cylinder sets[J]. J Lond Math Soc, 1945, 20: 110-120.

[11] Hu X Y, Taylor S J. Fractal properties of products and projections of measures in R^d[J]. Math Proc Camb Phil Soc, 1994, 115: 527-544.

[12] Hu X Y. Some fractal sets determined by stable processes[J]. Probab Theory Related Fields, 1994, 100: 205-225.

[13] 吴娟. 两条相互独立的非对称 Cauchy 过程轨道的乘积集的分形性质[J]. 数学杂志, 2000, 20: 63-70.

[14] Hou Y Y, Zhao M Z. Product fractal sets determined by stable processes[J]. Bulletin of the Australian Mathematical Society, 2009, 79: 201-212.

[15] Liang L Y. Exact Hausdorff measure for the product set of the ranges of stable component processes[J]. Journal of University of Chinese Academy of Sciences, 2014, 31(4): 453-459(in Chinese).梁龙跃.稳定分量过程像集乘积集的确切 Hausdorff 测度[J].中国科学院大学学报, 2014, 31(4):453-459.

[16] Ehm W. Sample function properties of the multi-parameter stable processes[J]. Z Wahrsch verw Geb 1981, 56: 195-228.

[17] 石海华. 多参数随机过程的样本轨道性质[D]. 北京: 中国科学院大学, 2012.

[18] Falconer K J. Fractal geometry: mathematical foundations and applications[M]. Wiley, 2007.

[19] Pruitt W E, Taylor S J. The potential kernel and hitting probabilities for the general stable process in R^N[J]. Trans Am Math Soc, 1969, 146: 299-321.

[20] Strook D W. Probability theory: an analytic view[M] 2nd ed(Rev). Cambridge: Cambridge University Press, 2010.