# Wold Decomposition -- summation to infinity

In Wold's decomposition we have $$Y_t = \sum_{j=0}^\infty b_j \varepsilon_{t-j} + \eta_t$$, where the variables have definition as in the Wikipedia page.

I'm confused about why the summation goes to infinity. What does this equation mean when $$j > t$$?

I think what the equation is trying to say is that the summation actually goes to $$t$$, but that we do this for all our $$t$$s (at least, that's what I gather from the answer to this question, where the summation of $$j$$ goes from $$-\infty$$ to $$\infty$$). Would it be correct to interpret the decomposition this way? If so, then the decomposition can be thought of as

$$Y_t = \sum_{j=0}^t b_j \varepsilon_{t-j} + \eta_t$$

but where we work this same equation through all our $$T \in \left(0,1,...,t \right)$$.

Is that right, or is there something I'm missing? Thank you.

• If you consider what it means for $Y_t$ to be second-order stationary, you can deduce what the $\varepsilon_{t-j}$ must be for $j\gt t.$
– whuber
Commented Oct 12, 2021 at 14:06
• Thanks as always whuber. I see it more clearly now. I was getting tripped up in the transition from the general form to actual implementation. Commented Oct 15, 2021 at 13:18