Stochastic Processes As Curves in Hilbert Space
Stochastic Processes As Curves in Hilbert Space
复制标题
希尔伯特空间中的随机过程曲线
DOI：
10.1137/1109032
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发表时间：
1964
影响因子：
0.6
通讯作者：
H. Cramér
中科院分区：
文献类型：
--
作者：
H. Cramér
利用Hilbert空间几何方法研究了正则复值二阶矩有限的随机过程x（t）。给出了过程x（t）的“过去和现在新息”的表示式（4）。数N被称为过程$x（t）$的完全谱重数，并且是存在这种表示的最小数。证明了x（t）的重数由相应的相关函数唯一确定，并且总是可以找到一个具有预先给定的重数的调和过程x（t）.
Regular complex-valued random processes $x(t)$ with finite moments of second order are studied by methods of Hilbert space geometry. A representation formula (4) is given for the process $x(t)$ in terms of “past and present innovations”. The number N is called the complete spectral multiplicity of the process $x(t)$ and is the smallest number for which such a representation exists. It is shown that the multiplicity of $x(t)$ is uniquely determined by the corresponding correlation function and that one can always find a harmonizing process $x(t)$ which has the multiplicity prescribed in advance.