determinisms of Set-Indexed Random Field and their Applications

集合索引随机场的确定性及其应用

• 批准号: 13640145
• 负责人: TAKENAKA Shigeo
• 金额: $ 0.7万
• 依托单位: Okayama University of Science
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640145/
• 关键词: stable random fields; random processes; determinsim; multi-dimensional time; 決定性; stable process; set Indexed fields; determinism

Random fields with parameter in subsets of a certain measure space are called Set-Indexed Random Fields and have a special property called determinism. A Gaussian system has 2-dimensional determinism, that is, the higher dimensional distributions of the system are completely determined by their own 1-dimensional and 2-dimensional (covariances) marginal distributions. On the contrary, there are some examples of non-Gaussian set-indexed random fields which are determined by their 3-dimensional marginals but not determined by 2-dimensional marginals.Let fix a convex cone V in the n-dimensional Euclidean space. A curve L is called time-like if for any point x in L, the "future" of the line is included in V+x, and the "past" is included in -V+x. An n-parameter random field X is called multi-parameter additive process if the restriction X|L on any time-like curve L is an additive process, that is X|L has independent and uniform increments. In this research, we obtained a characterization of n-parameter additie processes as set-indexed processes.Also, a characterization of 1-parameter processes valued in the space of time-like curves is obtained.

参数在某个测度空间的子集中的随机场称为集指标随机场，它具有一个特殊的性质，称为确定性。高斯系统具有二维决定性，也就是说，系统的高维分布完全由它们自己的一维和二维（协方差）边缘分布决定。相反，也有一些非高斯集指标随机场由其3维边值决定而不由2维边值决定的例子。曲线L称为类时曲线，如果对于L中的任何点x，该线的“未来”包含在V+x中，而“过去”包含在-V+x中。一个n指标随机场X称为多指标可加过程，如果约束X|在任何类时曲线上L是一个可加过程，即X| L具有独立的和均匀的增量。本文给出了n-指标可加过程作为集指标过程的一个刻划，同时也给出了值在类时曲线空间中的1-指标过程的一个刻划。

项目成果

J.K.Misiewicz, S.Takenaka: "Some remarks on SαS, β-substable random vectors"Probability and Mathematical Statistics. (印刷中). 7 (2003)

J.K.Misiewicz、S.Takenaka：“关于 SαS、β-亚稳定随机向量的一些评论”《概率与数学统计》（出版中）。

S.Takenaka: "Linearly additive random fields with independent increments on time-like curves"Probability and Mathematical Statistics. 受理. 7 (2003)

S.Takenaka：“类时间曲线上具有独立增量的线性加性随机场”概率与数学统计 7 (2003)。

J.K.Misiewicz., and S.Takenaka.: "Some remarks on SαS, β-substable random vectors"Probability and Mathematical Statistics, Now printing, 7pages.

J.K.Misiewicz. 和 S.Takenaka.：“关于 SαS、β-亚稳定随机向量的一些评论”《概率与数理统计》，正在印刷，7 页。

竹中茂夫: "Set-Indexed Processに関する話題"統計数理研究所レポート. 137. 61-71 (2001)

Shigeo Takenaka：“集合索引过程的主题”统计数学研究所报告 137. 61-71 (2001)。

高嶋 惠三: "擬似乱数とRandom Walk検定"神戸大学、Rokko Lectures in Mathematics 11. 42 (2002)

高岛敬三：“伪随机数和随机游走测试”神户大学，六甲数学讲座 11. 42 (2002)