| Hausdorff-type Measures of the Sample Path of Gaussian Random Fields |
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| 引用本文： | Zhen-long Chen San-yang Liu. Hausdorff-type Measures of the Sample Path of Gaussian Random Fields[J]. 应用数学学报(英文版), 2005, 21(4): 623-636. DOI: 10.1007/s10255-005-0270-7 |
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| 作者姓名： | Zhen-long Chen San-yang Liu |
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| 作者单位： | [1]School of Statistics and Computation Science, Zhejiang Gongshang University, Hangzhou 310035, China [2]Department of Applied Mathematics, Xidian University, Xi＇an 710071, China |
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| 基金项目： | Supported by the National Natural Science Foundation of China （No.10471148）, Sci-tech Innovation Item for Excellent Young and Middle-Aged University Teachers and Major Item of Educational Department of Hubei （No.2003A005）Acknowledgements. We wish to express our sincere thanks to Professor Xiao Yimin for suggesting the problem to me and for his subsequent encouragement and help. |
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| 摘 要： | Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.
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| 关 键 词： | Hausdorff测量 高斯随机域 图表集 测量函数 |
| 收稿时间： | 2004-10-20 |
| 修稿时间： | 2004-10-202005-03-08 |
Hausdorff-type Measures of the Sample Path of Gaussian Random Fields |
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| Zhen-long Chen,San-yang Liu. Hausdorff-type Measures of the Sample Path of Gaussian Random Fields[J]. Acta Mathematicae Applicatae Sinica, 2005, 21(4): 623-636. DOI: 10.1007/s10255-005-0270-7 |
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| Authors: | Zhen-long Chen San-yang Liu |
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| Affiliation: | (1) School of Statistics and Computation Science, Zhejiang Gongshang University, Hangzhou 310035, China;(2) Department of Applied Mathematics, Xidian University, Xi'an 710071, China |
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| Abstract: | Abstract Let ϕ be a Hausdorff measure function and Λ be an infinite increasing sequence of positive integers. The Hausdorff-type measure ϕ—m Λ associated to ϕ and Λ is studied. Let X(t)(t ∈R N ) be certain Gaussian random fields in R d . We give the exact Hausdorff measure of the graph set G r X([0, 1] N ), and evaluate the exact ϕ—m Λ measure of the image and graph set of X(t). A necessary and sufficient condition on the sequence Λ is given so that the usual Hausdorff measure function for X([0, 1] N ) and G r X([0, 1] N ) are still the correct measure functions. If the sequence Λ increases faster, then some smaller measure functions will give positive and finite (ϕ, Λ)-Hausdorff measure for X([0, 1] N ) and G r X([0, 1] N ). Supported by the National Natural Science Foundation of China (No.10471148), Sci-tech Innovation Item for Excellent Young and Middle-Aged University Teachers and Major Item of Educational Department of Hubei (No.2003A005) |
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| Keywords: | Gaussian random fields image graph Hausdorff-type measure |
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