Linear predictors from noise

Starting from an $ARMA$ process $y(t)=W(z)e(t)=\frac{C(z)}{A(z)}e(t)$, wehere $e(t) \sim WN(0,\lambda^2)$ we know that $y(t)$ is a steady state solution and $y(t) \sim MA(\infty)$, so: $$y(t) = w_0e(t)+w_1e(t-1)+...+w_ie(t-i)+...=\sum_{i=0}^{+\infty}{w_ie(t-i)}$$

Where $w_i = f(\text{parameters of }C(z)\text{ and }A(z))$.

The predictor: $\hat{y}(t+k|t) = a_0y(t)+a_1y(t-1)+...+a_iy(t-i)+...$ can be written as: $$\hat{y}(t+k|t) = a_0[\sum_{i=0}^{+\infty}{w_ie(t-i)}] + a_1[\sum_{i=0}^{+\infty}{w_ie(t-1-i)}] + ... =$$ $$= a_0[w_0e(t)+w_1e(t-1)+...]+a_1[w_0e(t-1)+w_1e(t-2)+...] + ...$$

Arranging the terms: $$\hat{y}(t+k|t) = \beta_0 e(t) + \beta_1 e(t-1) + ... = \sum_{i=0}^{+\infty}{\beta_ie(t-i)}$$ Where: $$ \begin{align} \beta_0 = a_0w_0 \newlinw \beta_1 = a_0w_1 + a_1w_1 ... \end{align} $$