We present two linear relations between an arbitrary (real tempered second order) generalized stochastic process over $\mathbb{R}^{d}$ and White Noise processes over $\mathbb{R}^{d}$. The first is that any generalized stochastic process can be obtained as a linear transformation of a White Noise. The second indicates that, under dimensional compatibility conditions, a generalized stochastic process can be linearly transformed into a White Noise. The arguments rely on the regularity theorem for tempered distributions, which is used to obtain a mean-square continuous stochastic process which is then expressed in a Karhunen-Lo\`eve expansion with respect to a convenient Hilbert space. The first linear relation obtained allows also to conclude that any generalized stochastic process has an orthogonal representation as a series expansion of deterministic tempered distributions weighted by uncorrelated random variables with summable variances. This representation is then used to conclude the second linear relation.
翻译:我们提出了一个任意的(真正温和的第二顺序)一般随机过程(在$\mathbb{R ⁇ d}$的基础上)与白色噪音过程(在$\mathb{R ⁇ d}$的基础上)之间的两个线性关系。第一个是,任何普遍的随机过程都可以作为白色噪音的线性转变获得。第二个是,在维度兼容条件下,一个普遍的随机过程可以线性地转换成白色噪音。这些论点依赖于温度分布的规律性理论,该理论用于获得一种平均平方连续的随机过程,然后以Karhunen-Lo ⁇ ⁇ eeve的扩展方式表达,以方便的希尔伯特空间为单位。获得的第一个线性关系还可以得出这样的结论,即任何一般的随机过程都具有某种或孔性的代表性,即以不相关随机变量加权的确定性温度分布成一系列扩展。然后,该表示用于得出第二个线性关系。