Understanding the infinite sum of random variables

Question

I am doing a course on time series analysis, and am struggling with this definition:

We call a weakly stationary process $\{X_t\}$ invertible with respect to a white noise $\{\epsilon_t\}$ if there exist real numbers $(\phi_j)_{j \in > \mathbb{N_0}}$ with $\sum_{j=0}^{\infty}|\phi_j| < \infty$ and $\epsilon_t = \sum_{j=1}^{\infty}\phi_j X_{t-j}$ for all $t \in > \mathbb{Z}$.

What exactly does it mean for a random variable, $\epsilon_t$ in this case, to be equal to an infinite series of random variables, $\sum_{j=1}^{\infty}\phi_j X_{t-j}$?

For some real sequence $\{x_t\}$, we write $L = \sum_{j=0}^{\infty}x_j$ if $\lim_{n \to \infty} s_n$ exists and equals $L$, where $s_n = \sum_{j=0}^{n} x_j$, but I cannot see how this definition extends to a random sequence?

Answer 1

Each of your random variables $X_t$ is a (real-valued) function defined on some probability space $(\Omega, \mathbb{B}, \mathbb{P})$, that is, $X_t \colon \Omega\mapsto\mathbb{R}$. So $\epsilon_t = \sum_{j=1}^{\infty}\phi_j X_{t-j}$ means that for all $\omega$ (in some subset of $\Omega$ with probability one) we have that $$ \sum_{j=1}^{\infty}\phi_j(\omega) \cdot X_{t-j}(\omega) $$ converges to a real number $\epsilon_t(\omega)$ in the usual sense of convergence of sequences of real numbers. Or, what I have described above (leaving out measurability issues) is the meaning of almost sure convergence (or convergence with probability one.) In the context of weak-sense stationary processes often one uses the concept of convergence in mean square in its place, which means that $$ \epsilon_t - \sum_{j=1}^{T}\phi_j X_{t-j} $$ have a variance that converges to zero when $T \to \infty$. A similar question with answer: Convergence of Sequence Random Variables