I have a question regarding the text's treatment of formula (4.1.1). It defines Y_t as depending on the previous t unobserved white noise terms. Therefore, Y_t is not an infinite series even though the author treats it as an infinite series.

In particular, formulas on page 56 take advantage of the geometric series formula 1/(1-x) = sum_{k=0}^{infinity} x^k. These sums cannot be infinite since the sum is counting down the e_k used from e_t to e_0. Shouldn't the author be using the partial sum formula

(1-r^{t+1})/(1-r) = sum_{k=0}^t r^k?

[NEAS: Module 9 discusses this topic more fully. For non-stationary time series, one needs a starting point; the textbook uses "-m". For stationary time series, we think of the series as infinite. If the time series is not very short, this is a good approximation, and it clarifies the equilibrium in the stationary time series.]

In a general linear process, time series attempts to capture "time" which (some may argue otherwise) does not have a starting point.  So there are indeed an infinite number of psi-weights and error terms prior to time t.

If t = now, then:

t-1 = one period ago,

t-2 = two periods ago,

t-3 = three periods ago,

...

t-1,000,000 = one million time periods ago,

... so on and so forth until the dawn of time, which in this mathematical sense doesn't exist.