Moving average processes

Given a zero mean white noise $e(t) \sim WN(0,\lambda^2)$ we define the moving average process of order $MA(n)$ as: $$y(t)=c_0e(t)+c_1e(t-1)+...+c_ne(t-n)$$

Which is a linear combination of the past values of $e(t)$ from $t$ to $t-n$ with real parameters $c_0$,$c_1$,...,$c_n$.

Weak description and stationarity

Given a moving average process $y(t) \sim MA(n)$ then we can easily demostrate that the mean function is always null: $$m_y(t)=\mathbb{E}[y(t)]=0$ because $e(t) \sim WN(0,\lambda^2$$

And the covariance function is: $$ \gamma_y(\tau)= \begin{cases} (c_0^2+c_1^2+...+c_n^2)\lambda^2 &\text{if } \tau=0 \newline (c_0c_1+c_1c_2+...+c_{n-1}c_n)\lambda^2 &\text{if } \tau=\pm1 \newline \text{...} \newline c_0c_n\lambda^2 &\text{if } \tau=\pm n \newline 0 &\text{if } |\tau|>n \end{cases} $$ Which is time invariant so $y(t)$ is stationary.