Computing the covariance function of a specific process

Find the covariance function of a Poisson process with constant intensity $\lambda$. Compare the answer obtained with the covariance function of the Wiener process.

Let $X = \left\{ X(t) = \xi e^{i(\eta t+\nu )}, t \in \mathbb {R}\right\}$, where the real random variables $\xi , \eta , \nu$ are such that $\nu$ is uniformly distributed on $[0,2 \pi ]$ and independent of $(\xi , \eta )$, and $\mathbb {E}\left[\xi^{2}\right] < \infty$. Verify that $X$ gives an example of a centered stationary process with covariance function of the form (36), where the measure $G = \mathbb {E}\left[\xi^{2}\right] F$, and $F$ is the probability distribution of the variable $\eta$. Now put $Y = \left\{ Y(t) = \xi \cos (\eta t+\nu ), t \in \mathbb {R}\right\}$. Check that $Y$ is a stationary process. If $X = \left\{ X(t), t \in \mathbb {R}\right\}$ is a stationary complex-valued process, can one assert that $\left\{ \operatorname {Re} X(t), t \in \mathbb {R}\right\}$ and $\left\{ \operatorname {Im} X(t), t \in \mathbb {R}\right\}$ are stationary processes?

Formula (36):

where $G = G(d \lambda )$ is some nonnegative finite ("spectral") measure on $\mathscr {B}(\mathbb {R})$.

Let $\eta$ be a random variable with distribution function $F(x)$. Find all finite-dimensional distributions of the random process $\xi (t) = \eta + t$, its expectation $m(t)$, and its covariance function $B(t, s)$.

Let $\eta$ be a random variable uniformly distributed on the interval $[0, 1]$. Find all finite-dimensional distributions of the random process $\xi (t) = \; \mathbb {1}_{t < \eta }$, $t \in [0, 1]$, its expectation $m(t)$, and its covariance function $B(t, s)$.

Let the random variable $\xi$ have a normal distribution with expectation $a$ and variance $\sigma^{2}$; let $b \in \mathbb {R}$. Find all finite-dimensional distributions of the random process $\xi (t) = \xi t + b$, its expectation $m(t)$, and its covariance function $B(t, s)$.

Let $\xi$ and $\eta$ be uncorrelated random variables with zero means and finite variances, and let $\alpha , \beta > 0$ be constants. Find the expectation $m(t)$ and the covariance function $B(t, s)$ of the random process $\xi (t) = \xi e^{-\alpha t} + \eta e^{-\beta t}$.

Let $\xi$ and $\eta$ be uncorrelated random variables with zero means and unit variances. Find the expectation $m(t)$ and the covariance function $B(t, s)$ of the random process $\xi (t) = t + \xi \cos \gamma t + \eta \sin \gamma t$.

Let a random process $\xi (t)$ equal $\eta \cos (\gamma t - \theta )$, where $\eta$ is a random variable with zero mean and variance $\sigma^{2}$; $\theta$ is a random variable uniformly distributed on the interval $[0, 2\pi ]$; $\gamma$ is a nonrandom parameter. The random variables $\eta$ and $\theta$ are independent. Find the expectation and the covariance function of the process $\xi (t)$. Under what conditions is $\xi (t)$ a stationary process?

Let $\xi (t) = \eta_{1}f_{1}(t) + \ldots + \eta_{n}f_{n}(t)$, where $\eta_{1}, \ldots , \eta_{n}$ are uncorrelated random variables with zero means and variances $\sigma_{1}^{2}, \ldots , \sigma_{n}^{2}$ respectively, and $f_{1}, \ldots , f_{n}$ are nonrandom functions. Find the expectation and the covariance function of the process $\xi (t)$.

Find the correlation function of the random process $\xi_{t} = \gamma_{1} f_{1}(t)+\ldots +\gamma_{n} f_{n}(t)$, where $f_{1}(t), \ldots , f_{n}(t)$ are nonrandom functions, and $\gamma_{1}, \ldots , \gamma_{n}$ are uncorrelated random variables with variances $d_{1}, \ldots , d_{n}$, respectively.

Let $\xi_{t}^{(1)}, \ldots , \xi_{t}^{(n)}$ be independent random processes such that $\mathbb {E}\left[\xi_{t}^{(i)}\right] = 0, i = 1, \ldots , n$; let $K_{1}(t, s), \ldots , K_{n}(t, s)$ be the corresponding correlation functions. Find the correlation function of the process $\xi_{t}^{(1)}+\ldots +\xi_{t}^{(n)}$.

Let $\xi$ be a random variable having a normal distribution with mean $m$ and variance $\sigma^{2}$, $b$ a real number. Find the correlation function of the process

Let $A, \eta$ and $\varphi$ be random variables, with $\varphi$ independent of $A$ and $\eta$, $A \geqslant 0, \eta \geqslant 0$, and $\varphi$ uniformly distributed on the segment $[0,2 \pi ]$. Find the mean and correlation function of the process

Let $\xi_{t}^{(1)}$ and $\xi_{t}^{(2)}$ be two independent random processes with correlation functions $K_{1}(t, s)$ and $K_{2}(t, s)$ respectively. Find the correlation function of the process

Let $w_{s}^{(1)}$ and $w_{s}^{(2)}$ be independent Wiener processes. For every real $t$, set

Further, let $\xi_{t} = \frac{1}{h}\left(\bar{w}_{t}-\bar{w}_{t-h}\right), \quad h = \text{const}$. Show that $\xi_{t}$ is a stationary process. Find its covariance function and spectral density.

Show that the covariance matrix $C = \left(t_{j} \wedge t_{k}\right)_{j, k = 1, \ldots , n}$ appearing in Theorem 2.6 is positive definite.

Find the correlation function $C(s, t): = \mathbb {E}\left[X_{s} X_{t}\right], s, t \geqslant 0$, of the stochastic process $X_{t}: = B_{t}^{2} - t, t \geqslant 0$, where $\left(B_{t}\right)_{t \geqslant 0}$ is a $\mathrm{BM}^{1}$.