# Understanding the infinite sum of random variables

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?

## 1 Answer

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

• Thank you so much kjetil. I had to read up on probability spaces and random variables before understanding your explanation, but now it makes sense. I just have a couple of follow up questions. First, what is an example of an outcome, $\omega$, if the probability space is used to model stock prices, for instance? Would it just be a possible stock price? In that case, would each random variable, $X_t$, be defined by the identity function? – Henry Apr 16 at 15:56