Almost sure convergence of linear processes

Question

In Brockwell and Davis's book (Introduction to time series and forecasting), a linear process is defined as

$ X_t = \sum_{j=-\infty}^{\infty} \psi_jZ_{t-j}$

where $Z_{t} \sim WN(0, \sigma^2)$, $\psi_j$'s are constants such that $\sum_{j=-\infty}^{\infty}|\psi_j| < \infty$.

It states that $\sum_{-\infty}^{\infty}|\psi_j| < \infty$ ensures that the infinite sum converges (with probability one) as $E|Z_t| \leq \sigma < \infty$ and $E|X_{t}| < \infty$.

I am not sure how these three conditions help in proving almost sure convergence of the infinite sum.

Answer 1

We have $$ \mathbb{E}\left[ \sum_{j=-\infty}^\infty |\psi_jZ_{t-j}| \right] = \sum_{j=-\infty}^\infty \mathbb{E}[|\psi_j|\cdot|Z_{t-j}|] = \sum_{j=-\infty}^\infty |\psi_j|\cdot\mathbb{E}[|Z_{t-j}|] \leq \sigma \sum_{j=-\infty}^\infty |\psi_j| < \infty. $$ This means that, for every $t$, the series defining $X_t$ is almost surely absolutely convergent, because the series computed above is almost surely finite.