论文标题
随机措施的Karhunen-Loève扩展
Karhunen-Loève expansion of Random Measures
论文作者
论文摘要
我们为$ \ mathbb {r}^{d} $带来的真实，功能调节的二阶随机度量的正交扩展，并进行了协方差。这样的扩展可以看作是karhunen-loève分解，包括一系列确定性的实际度量，由不相关的真实随机变量加权，而方差形成融合序列。该系列的收敛性在随机方面是在均值的意义上，并且在衡量意义上反对可测量的有界测试函数（如果随机度量不是有限的，则具有紧凑的支持），这意味着设定的收敛性。事实证明，这是利用额外的要求，即在$ \ mathbb {r}^{d} \ times \ times \ mathbb {r}^{d} $上，描述随机度量的协方差结构，为此我们还提供系列扩展。这些结果涵盖了例如高斯白噪声，泊松和Cox点过程的情况，可用于获得拖网过程的扩展。
We present an orthogonal expansion for real, function-regulated, second-order random measures over $\mathbb{R}^{d}$ with measure covariance. Such a expansion, which can be seen as a Karhunen-Loève decomposition, consists in a series of deterministic real measures weighted by uncorrelated real random variables with the variances forming a convergent series. The convergence of the series is in a mean-square sense stochastically and against measurable bounded test functions (with compact support if the random measure is not finite) in the measure sense, which implies set-wise convergence. This is proven taking advantage of the extra requirement of having a covariance measure over $\mathbb{R}^{d}\times\mathbb{R}^{d}$ describing the covariance structure of the random measure, for which we also provide a series expansion. These results cover for instance the cases of Gaussian White Noise, Poisson and Cox point processes, and can be used to obtain expansions for trawl processes.
