论文标题
通过不当线性功能对高斯过程的随机建模
Stochastic modelling of Gaussian processes by improper linear functionals
论文作者
论文摘要
存在各种随机过程的方法，并指出诸如可测量性和连续性之类的关键特性并不能够轻松满足。我们介绍了使用不当线性功能的高斯过程的新理论。使用I.I.D.的集合标准正常变量，我们定义高斯白噪声并讨论其特性。这将扩展到希尔伯特空间上的高斯一般过程，在那里允许方差为任何合适的操作员。我们的主要重点是$ l^2 $空间，我们讨论了在这种情况下连续的高斯流程的标准。最后，我们概述了使用介绍的理论的统计推断框架，重点是$ l^2 [0,1] $。我们将Fredholm的决定因素引入功能对数可能性。我们证明，天真的功能对数可能性与多元可能性不一致。引入了校正项，我们证明了一个渐近结果。
Various approaches to stochastic processes exist, noting that key properties such as measurability and continuity are not trivially satisfied. We introduce a new theory for Gaussian processes using improper linear functionals. Using a collection of i.i.d. standard normal variables, we define Gaussian white noise and discuss its properties. This is extended to general Gaussian processes on Hilbert space, where the variance is allowed to be any suitable operator. Our main focus is $L^2$ spaces, and we discuss criteria for Gaussian processes to be continuous in this setting. Finally, we outline a framework for statistical inference using the presented theory with focus on the special case of $L^2[0,1]$. We introduce the Fredholm determinant into the functional log-likelihood. We demonstrate that the naive functional log-likelihood is not consistent with the multivariate likelihood. A correction term is introduced, and we prove an asymptotical result.
