这是一份 Imperial帝国理工大学 MATH97108作业代写的成功案例
Moments of a time series. Assuming they exist, we define the mean function $\mu(t)$ and autocovariance function $\gamma(t, s)$ of $\left(X_{t}\right){t \in \mathbb{Z}}$ by $$ \begin{aligned} \mu(t) &=E\left(X{t}\right), & & t \in \mathbb{Z}, \
\gamma(t, s) &=E\left(\left(X_{t}-\mu(t)\right)\left(X_{s}-\mu(s)\right)\right), & t, s \in \mathbb{Z} .
\end{aligned}
$$
It follows that the autocovariance function satisfies $\gamma(t, s)=\gamma(s, t)$ for all $t, s$, and $\gamma(t, t)=\operatorname{var}\left(X_{t}\right) .$
(covariance stationarity). The time series $\left(X_{t}\right)_{t \in \mathbb{Z}}$ is covariance stationary (or weakly or second-order stationary) if the first two moments exist and satisfy
$$
\begin{aligned}
\mu(t) &=\mu, & & t \in \mathbb{Z} \
\gamma(t, s) &=\gamma(t+k, s+k), & t, s, k \in \mathbb{Z}
\end{aligned}
$$
MATH97108 COURSE NOTES ：
For all practical purposes we can restrict our study of ARMA processes to causal ARMA processes. By this we mean processes satisfying which have a representation of the form
$$
X_{t}=\sum_{i=0}^{\infty} \psi_{i} \varepsilon_{t-i},
$$
where the $\psi_{i}$ are coefficients which must satisfy
$$
\sum_{i=0}^{\infty}\left|\psi_{i}\right|<\infty .
$$