Time Series: stationary AR(1) -> MA(infinity)

[kingwinner]

Theorem:
A stationary AR(1) model can be expressed in terms of MA(infinity).

Proof:

Now I don't understand how they get from the second last line to the last line. Where did the term Y_{t-m} go?

I understand you can keep doing the substitution iteratively, but you always have to end up with a "Y" term no matter how many times you do it, but on the last line of the proof there is no "Y" term, so it magically disappared? I'm really confused.

Hopefully someone can explain this (in simple terms if possible). Thank you!

 

[tkhunny]

Does "infinity" actually mean "unbounded iterations" or does it mean "until you can take no more differences"?

 

[kingwinner]

No idea. But I believe an infinite series should be a limit.

I think it's the limit as m->?.

As m->?,
?[sup:1xh74vlo]m[/sup:1xh74vlo] ->0
BUT you also have to look at what happens to the other term, right? What if Y[sub:1xh74vlo]t-m[/sub:1xh74vlo]->?? Then we have indeterminate form 0*?, and we just can't conclude that the product->0. So I just really don't understand how the Y term magically disappears...