\begin{problem}{1}
Consider a process of the form
$$Y_t = R \cos  [2\pi(ft + \Phi)], \quad t = 0; \pm1;\pm 2 \dots, $$
where $0<f< 1/2$ is deterministic, $R$ and $\Phi$ are uncorrelated random variables with $\Phi$ being uniformly distributed in [0, 1].
\begin{enumerate}[label=\alph*)]
    \item Compute expectation, autocovariance function and autocorrelation function of ($Y_t$)
    \item Check whether the process $Y$ is stationary (in a weak sense).
\end{enumerate}
\end{problem}

\begin{problem}{2}
Let
$$X_t = \varphi X_{t-3} + \varepsilon_t,$$
where ($\varepsilon_t$) is a white noise with $\varepsilon_t ~ N(0;\sigma^2)$.

\begin{enumerate}[label=\alph*)]
    \item For which values of $\varphi$ is the process ($X_t$) weakly stationary?
    \item Find the autocorrelation function and the spectral density of ($X_t$).
\end{enumerate}
\end{problem}

\begin{problem}{3}
It is known that a time series $(X_t)_{t \in \mathbb{Z}}$ is stationary with the spectral density of the form

$$ S(\lambda) = \frac{1}{\pi}\cos^2 (\lambda)(1+cos(\lambda)), -\pi<\lambda<\pi.$$

Find the autocorrelation function of ($X_t$).
\end{problem}

\begin{problem}{4}
Define a process $(Y_t)_{t \in \mathbb{Z}}$ via
$$Y_t = X_t + 0.4 X_{t-1} + 0.04 X_{t-2},$$
where ($X_t$) is a stationary AR(1) process with the autocorrelation function $C_X (h) = 0,5^{|h|}$.

\begin{enumerate}[label=\alph*)]
    \item Is $(Y_t)_{t \in \mathbb{Z}}$ (weakly) stationary ?
    \item Compute the autocorrelation function of ($Y_t$).
    \item Compute the spectral density of ($Y_t$).
\end{enumerate}
\end{problem}

\begin{problem}{5}
Let a function $\ro: \mathbb{Z} \implies \mathbb{R}$ is given by
$$\ro(h) = 1.5(h=0) + 0.75(|h| = 1) + 0.5(|h| = 2).$$
Is $\ro$ an autocovariance function of some stationary process?
\end{problem}

\begin{problem}{6}
Consider ARMA(1,1) process of the form
$$X_t = 0.4X_{t-1} + \varepsilon_t + 0.25\varepsilon_{t-1},$$
where $\varepsilon_t ~ N(0;\sigma^2)$ is a white noise.

\begin{enumerate}[label=\alph*)]
    \item Check whether the process ($Y_t$) is stationary and invertible.
    \item Compute the spectral density of ($X_t$).
\end{enumerate}

\end{problem}