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?