L2 and a.s. ergodic theorem for stationary sequences via covariance decay

Let $\left\{ X_{n}, n \in \mathbb {Z}\right\}$ be a stationary process with mean $a$ and covariance function $R = R(n), n \in \mathbb {Z}$. Prove that as $N \rightarrow \infty$

Suppose that, under the conditions of the preceding exercise,

for some $c, \gamma > 0$ and all $N \in \mathbb {N}$. Prove that then

Show that condition (85) holds if $R(n) = O\left(n^{-\gamma }\right)$ as $n \rightarrow \infty$.

Let $X_{1}, X_{2}, \ldots$ be a real-valued stationary sequence with zero means and autocovariance function $c(m)$. Show that

where $F$ is the spectral distribution function. Deduce that $n^{-1} \sum_{j = 1}^{n} X_{j} \xrightarrow {\text{ m.s. }} 0$ if and only if $F(0)-F(0-) = 0$, and show that

Assume that $X_{1}, X_{2}, \ldots$ is a stationary sequence with autocovariance function $c(m)$. Show that

Assuming that $j^{-1} \sum_{i = 0}^{j-1} c(i) \rightarrow \sigma^{2}$ as $j \rightarrow \infty$, show that

Let $X = \left\{ X_{n}: n \geq 1\right\}$ be stationary with constant mean $\mu = \mathbb {E}\left[X_{n}\right]$ for all $n$, and such that $\operatorname {Cov}\left[X_{1}, X_{n}\right] \rightarrow 0$ as $n \rightarrow \infty$. Show that $n^{-1} \sum_{j = 1}^{n} X_{j} \xrightarrow {\text{ m.s. }} \mu$.

Let $f_1,\ldots ,f_n$ be different (non-random) frequencies, and let $A_1,\ldots ,A_n$ and $\phi_1,\ldots ,\phi_n$ be mutually independent random variables, with all $\phi_k$ uniform on $[0,2\pi ]$. Let $A_0$ be a random constant, independent of all other $A_k$ and $\phi_k$. Consider the process

and the time-averaged power $E_T=\frac{1}{T}\int_0^T X(t)^2\, \mathrm{d}t$ over the interval $[0,T]$. Expanding $X(t)^2$ by trigonometric identities gives

It is stated to be "easy to see" that $T^{-1}\int_0^T\cos (2\pi f_k t+\phi_k)\, \mathrm{d}t \to 0$ as $T\to \infty$, and also, for $f_k \neq f_l$,

so that, as $T\to \infty$,

Complete the "easy to see" part of this convergence, i.e., prove the three limit statements above.

Let $Y$ and $\left\{ X_n, n=1,2,\ldots \right\}$ be independent normal variables and define $Z_n = X_n+Y$. Is $\left\{ Z_n, n=1,2,\ldots \right\}$ a linearly ergodic sequence? Compare with the covariance criterion for ergodicity.

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

its spectral representation. Prove that

Let $\left\{ \xi_{k}\right\}$ be a stationary sequence, $\mathbb {E}\left[\xi_{0} \xi_{n}\right]-\left(\mathbb {E}\left[\xi_{0}\right]\right)^{2} \rightarrow 0$ as $n \rightarrow \infty$. Prove that

Let the correlation function of a random process $\xi_{t}$ stationary in the wide sense tend to zero at infinity. Prove that $\frac{1}{t_{2}-t_{1}} \int_{t_{1}}^{t_{2}} \xi_{t} d t$ converges to $\mathbb {E}\left[\xi_{t}\right]$ in the mean square as $t_{2}-t_{1} \rightarrow \infty$.