determinisms of Set-Indexed Random Field and their Applications

在一定测度空间的子集中具有参数的随机场称为集索引随机场，它具有确定性的特殊性质。高斯系统具有二维决定论，即系统的高维分布完全由其自身的一维和二维（协方差）边际分布决定。相反，有一些非高斯集索引随机场的例子，它们是由它们的三维边际决定的，而不是由二维边际决定的。在n维欧氏空间中固定一个凸锥V。曲线L被称为类时曲线，如果对于L中的任何点x，线的“未来”包含在V+x中，而“过去”包含在-V+x中。如果任意类时曲线L上的约束X|L是一个加性过程，即X|L具有独立且均匀的增量，则n参数随机场X称为多参数加性过程。在本研究中，我们得到了n参数可加过程作为集索引过程的一个表征。此外，还得到了在类时曲线空间中取值的1参数过程的表征。

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.