Stationary random distributions
Stationary random distributions
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平稳随机分布
DOI：
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发表时间：
1954
期刊：
影响因子：
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通讯作者：
Kiyosi Itô
中科院分区：
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作者：
Kiyosi Itô
与L的分布概念相同。Schwartz [1111]作为函数的一种推广而引入，我们可以将平稳随机分布定义为平稳随机函数的一种推广，即平稳过程。这种考虑将使我们能够建立平稳过程、布朗运动过程、平稳增量过程和其它类似过程的统一理论。我们将首先在§ 1中介绍一些基本概念。在§ 2中，我们将定义平稳随机分布的协方差分布，它对应于Khintchine的协方差函数[7]。在§ 3和§ 4中，我们将分别证明协方差分布和平稳随机分布的谱分解定理。在§ 5中，我们将讨论平稳分布的导数。在§ 6中，我们将证明，任何平稳分布都可以用具有平稳k阶增量的某个连续过程的k阶导数（在分布的意义上）来识别，对于某个k。
In the same way as the concept of distributions by L. Schwartz [1111 ) was introduced as an extended one of functions, we may define stationary random distributions as an extension of stationary random functions viz, stationary processes. Such consideration will enable u s to establish a unified theory of stationary processes, Brownian motion processes, processes with stationary increments and other similar ones, a s is shown in this p a p e r . We shall first introduce some fundamental notions in § 1. In § 2 we shall define the covariance distribution of stationary random distributions, which corresponds to Khintchine's covariance function [7]. I n § 3 a n d § 4 we shall prove the spectral decomposition theorems of covariance distributions and stationary random distributions respectively. In § 5 we shall discuss the derivatives of stationary distributions. In § 6 we shall show that any stationary distribution is identified with a k-th derivative ( in th e sense of distributions) of a certain continuous process with stationary k-th order increments for some k.