determinisms of Set-Indexed Random Field and their Applications

参数在某一度量空间的子集上的随机场称为集索引随机场，具有确定性的特殊性质。高斯系统具有二维确定性，即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，也有一些非高斯集索引随机场的例子，它们是由它们的三维边缘决定的，而不是由二维边缘决定的。一条L曲线称为似时过程，如果对L中的任一点x，直线的“未来”包含在Vx中，而“过去”包含在-Vx中。如果任意一条似时曲线L上的约束X|L是一个可加过程，即X|L有独立且一致的增量，则称n指标随机场X为多参数可加过程。在这项研究中，我们得到了n指标加法过程是集指标过程的刻画，也得到了取值于类时曲线空间中的1指标过程的刻画。

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.