(a) Suppose $K(\cdot)$ is a complex valued function on the integers such that $\sum_{n=-\infty}^{\infty}|K(n)|<\infty$. Define
$$
f(\lambda)=\frac{1}{2 \pi} \sum_{n=-\infty}^{\infty} e^{-i n \lambda} K(n)
$$
and show that
$$
K(h)=\int_{-\pi}^\pi e^{i h x} f(x) d x, \quad h=0, \pm 1, \pm 2, \ldots
$$
(b) Let $\left\{\left\{X_n, n=0, \pm 1, \pm 2, \ldots\right\}\right.$ be a zero mean weakly stationary process. This means $E\left(X_m\right)=0$ for all $m$ and
$$
\gamma(h)=E\left(X_m X_{m+h}\right)
$$
is independent of $m$. The function $\gamma$ is called the autocovariance (acf) function of the process $\left\{X_n\right\}$.
Prove the following: An absolutely summable complex valued function $\gamma(\cdot)$ defined on the integers is the autocovariance function of a weakly stationary process iff
$$
f(\lambda)=\frac{1}{2 \pi} \sum_{n=-\infty}^{\infty} e^{-i n \lambda} \gamma(n) \geq 0, \text { for all } \lambda \in[-\pi, \pi],
$$
in which case
$$
\gamma(h)=\int_{-\pi}^\pi e^{i h x} f(x) d x
$$
(So $\gamma(\cdot)$ is a chf.)
Hint: If $\gamma(\cdot)$ is an acf, check that
$$
f_N(\lambda)=\frac{1}{2 \pi N} \sum_{r, s=1}^N e^{-i r \lambda} \gamma(r-s) e^{i s \lambda} \geq 0
$$
and $f_N(\lambda) \rightarrow f(\lambda)$ as $N \rightarrow \infty$. Use (9.34). Conversely, if $\gamma(\cdot)$ is absolutely summable, use (9.34) to write $\gamma$ as a Fourier transform or chf of $f$.
Check that this makes $\gamma$ non-negative definite and thus there is a Gaussian process with this $\gamma$ as its acf.
(c) Suppose
$$
X_n=\sum_{i=0}^q \theta_i Z_{n-i},
$$
where $\left\{Z_n\right\}$ are iid $N(0,1)$ random variables. Compute $\gamma(h)$ and $f(\lambda)$.
(d) Suppose $\left\{X_n\right\}$ and $\left\{Y_n\right\}$ are two uncorrelated processes (which means $E\left(X_m Y_n\right)=0$ for all $m, n$, and that each has absolutely summable acfs. Compute $\gamma(h)$ and $f(\lambda)$ for $\left\{X_n+Y_n\right\}$.