Spectral Representation

Let $\Lambda$ be a set, $\mathcal{A}$ an algebra of its subsets, and $\mu$ a measure on $\mathcal{A}$. Suppose the map $Z: \mathcal{A} \rightarrow L^{2}(\Omega , \mathcal{F}, \mathbb {P})$ satisfies the equality

Prove that $Z$ is an orthogonal measure on $\mathcal{A}$.

Let $\lambda_{1}, \ldots , \lambda_{k}$ be numbers from the interval $[-\pi , \pi ]$, and let $\Phi_{1}, \ldots , \Phi_{k}$ be centered, pairwise uncorrelated random variables. Prove that the process $\left(X_{n}, n \in \mathbb {Z}\right)$, where

is wide-sense stationary, and find its spectral representation.

Let $\left\{ X_{n}, n \in \mathbb {Z}\right\}$ be a wide-sense stationary sequence with mean $a$ and covariance function $R(n)$. Prove that

if and only if

Let the centered, wide-sense stationary process $\left(X_{n}, n \in \mathbb {Z}\right)$ satisfy the equality $X_{0}=X_{N}$ a.s. for some $N \in \mathbb {N}$. Prove that then for all $n \in \mathbb {Z}$ $X_{n}$ has the form

where $\Phi_{k}$ are centered, pairwise uncorrelated random variables from $L^{2}$.

Let $P(x)=a_{0}+a_{1} x+\ldots +a_{n} x^{n}$ be a polynomial, and let $P\left(\frac{d}{d t}\right)$ be the corresponding differentiation operator in $L^{2}$. Let $\left(\xi_{t}, t \in \mathbb {R}\right)$ be a wide-sense stationary process with known spectral representation. The wide-sense stationary process $\left(X_{t}, t \in \mathbb {R}\right)$ has a spectral representation and, moreover, satisfies the equation

Find the spectral representation for $X_{t}$. Under what conditions on the polynomial $P$ is the solution of the equation unique?

The stationary process $\left(Y_{t}, t \in \mathbb {R}\right)$ satisfies the equality $d Y_{t} / d t=X_{t}$, where the stationary process $\left(X_{t}, t \in \mathbb {R}\right)$ has spectral density $f(\lambda )=\lambda^{2} I\left\{ \left|\lambda \right|<1\right\}$. Find $\operatorname {cov}\left(Y_{1}, Y_{0}\right)$

The random process $\left(X_{t}, t \in \mathbb {R}\right)$ is centered and wide-sense stationary. Its spectral density equals $f(\lambda )=\left|\lambda \right| I_{[-2,2]}(\lambda )$. Using the spectral representation, find the spectral density of the process $Y_{t}$ satisfying the equation $\frac{d^{2}}{d t^{2}} Y_{t}+5 Y_{t}=X_{t}$. Compute D $Y_{1}$.

Let $\left(X_{t}, t \in \mathbb {R}\right)$ be a wide-sense stationary process, and

its spectral representation. Prove that