Hi All: This question is kind of an add on question to the gorgeous answer by Juancho provided at this link:
The answer explains how the limit of exponential smoothing given an impulse response $x_0$, (so that new $x_i$'s are not arriving. only $x_0$ arrives) can be computed by convolving the impulse with a step signal.
The difference between the framework at that link and my framework is that actual series is not driven by an impulse response. By this I mean that the same process, occurs over and over because a "new" $x_i$ is arriving during each period so one could think of there being an impulse response as described but occurring in each period. So, there is a step signal convolved with the first impulse response. Then, in the next period, another step signal is convolved with a second impulse, etc, etc. In other words, now there is $x_1$, $x_2$, $x_3$, ..... $x_n$, where each $x_i$ gets operated on in the same manner as described by Juancho. Assuming that my setup is clear, my question is: is there a simple recursion for this non-impulse response case. Does my case have a name ? It seems like there should be a way to use recursion but I can't figure it out. Thank you very much for any references, answers, books, whatever.