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 $c_0$,$c_1$,...,$c_n$.

MA(n) weak characterization and stationarity

Given a moving average process $y(t) \sim MA(n)$ then we can easily demostrate that the mean function is always null:

$$ m_y(t)=\mathbb{E}[y(t)]=\mathbb{E}[c_0e(t)+c_1e(t-1)+...+c_ne(t-n)]=\newline =c_0\mathbb{E}[e(t)]+c_1\mathbb{E}[e(t-1)]+...+c_n\mathbb{E}[e(t-n)]=0 \text{ because } e(t) \sim WN(0,\lambda^2) $$

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

MA(inf) process

The class of models $MA(\infty)$ is a huge class which covers almost all SSP.

Considering a zero mean white noise $e(t) \sim WN(0,\lambda^2)$ the process $y(t) \sim MA(\infty)$ is: $$y(t)=c_0e(t)+c_1e(t-1)+...+c_ie(t-i)+...=\sum_{i=0}^{+\infty}c_ie(t-i)$$ Which is an infinite sequence of random variables and the convergence to a proper random variance is not always guaranteed, only if: $$\sum_{i=0}^{+\infty}c_i^2 < +\infty$$

A moving average process of order $n$ is equal to a $MA(1infty)$ process with $c_i=0$ for $i>n$.

MA(inf) weak characterization and stationarity

The mean function of a well defined $MA(\infty)$ process is: $$m_y(t)=\mathbb{E}[y(t)]=\mathbb{E}[\sum_{i=0}^{+\infty}c_ie(t-i)]=0$$

While the covariance function: $$ \gamma(t,\tau)=\mathbb{E}[(y(t)-m_y(t))(y(t-\tau)-m_y(t-\tau))]=\newline =\mathbb{E}[y(t)-y(t-\tau)]=\mathbb{E}[\sum_{i=0}^{+\infty}c_ie(t-i)\sum_{j=0}^{+\infty}c_je(t-\tau-j)]=\newline [\text{Using linearity (well defined proces)}]\newline =\sum_{i,j=0}^{+\infty}c_ic_j\mathbb{E}[e(t-i)e(t-\tau-j)] $$ Since $\mathbb{E}[e(t-i)e(t-\tau-j)]=\gamma_e(t-i-(t-\tau-j))=\gamma_e(\tau+j-i)$ wich is always null except $\gamma_e(\tau+j-i)=\lambda^2$ for $i=j+\tau$. So: $$\gamma(t,\tau)=\lambda^2\sum_{j=0}^{+\infty}c_jc_{j+\tau}$$

Which is time invariant so the process is stationary.

Note that for $\tau=0$ we obtain the variance $\gamma_y(0)=\lambda^2\sum_{j=0}^{+\infty}c_j^2$ and the conditions of convergence makes the process well defined because if $\sum_{i=0}^{+\infty}c_i^2 < +\infty$ then $\gamma_{y}(0) < +\infty$ and since the process is stationary then the covariance is bounded by the variance, this implies that $\gamma_y(\tau)$ is well defined for every $\tau$.