Introduction

Let $\ldots , Z_{-1}, Z_{0}, Z_{1}, Z_{2}, \ldots$ be independent real random variables with means 0 and variances $1$, and let $\alpha , \beta \in \mathbb {R}$. Show that there exists a (weakly) stationary sequence $\left\{ W_{n}\right\}$ satisfying $W_{n} = \alpha W_{n-1}+\beta W_{n-2}+Z_{n}, n = \ldots ,-1,0,1, \ldots$, if the (possibly complex) zeros of the quadratic equation $z^{2}-\alpha z-\beta = 0$ are smaller than 1 in absolute value.

Let $U$ be uniformly distributed on $[0,1]$ with binary expansion $U = \sum_{i = 1}^{\infty } X_{i} 2^{-i}$. Show that the sequence

is strongly stationary, and calculate its autocovariance function.

Let $\left\{ X_{n}: n = \ldots ,-1,0,1, \ldots \right\}$ be a stationary real sequence with mean 0 and autocovariance function $c(m)$.

(i) Show that the infinite series $\sum_{n = 0}^{\infty } a_{n} X_{n}$ converges almost surely, and in mean square, whenever $\sum_{n = 0}^{\infty }\left|a_{n}\right| < \infty$.

(ii) Let

where $\sum_{k = 0}^{\infty }\left|a_{k}\right| < \infty$. Find an expression for the autocovariance function $c_{Y}$ of $Y$, and show that

Let $X = \left\{ X_{n}: n \geq 0\right\}$ be a discrete-time Markov chain with countable state space $S$ and stationary distribution $\mathbf{\pi }$, and suppose that $X_{0}$ has distribution $\mathbf{\pi }$. Show that the sequence $\left\{ f\left(X_{n}\right): n \geq 0\right\}$ is strongly stationary for any function $f: S \rightarrow \mathbb {R}$.

Let $W, X, Y, Z$ have a multivariate normal distribution with zero means and covariance matrix

Let $U = W+i X$ and $V = Y+i Z$. Show that $U$ and $V$ are uncorrelated but not independent. Why does this not violate the conclusion of Example (4.5.9) that multivariate normal random variables are independent if and only if they are uncorrelated?

Let $X_{n} = \cos (n S+U)$ for $n \in \mathbb {Z}$, where $U$ is uniformly distributed on $(-\pi , \pi )$, and $S$ is independent of $U$ with a density function $g$ that is symmetric on its support $(-\pi , \pi )$. Show that $X = \left\{ X_{n}: n \in \mathbb {Z}\right\}$ is weakly stationary, and find its autocorrelation function $\rho_{X}$.

Let $Y_{n} = Z_{n}+a Z_{n-1}$ for $n \in \mathbb {Z}$, where $\left|a\right| < 1$ and the $Z_{n}$ are independent, identically distributed random variables with means 0 and variances 1. Is it possible to choose $g$ in such a way that $Y = \left\{ Y_{n}: n \in \mathbb {Z}\right\}$ has the same autocorrelation function as $X$? If so, how?