Say I have a time series of observations and I compute a measure of the variance of that time series as the standard deviation (SD) in a rolling window of width $w$ and that window is moved in single time steps over the series. Assume further that $w = \left \lceil{n/2}\right \rceil$, where $n$ is the number of observations, and that the window is right aligned; I have to observe $w = \left \lceil{n/2}\right \rceil$ values of the series before I start yielding moving window estimates of the SD of the time series.
Is there an expected form for the ACF of the new time series of SD values? I presume the dependence on previous values will relate to the window with $w$, but is the ACF of such a series related to the ACF of an $\mathrm{MA}(w)$ process?
Background
I'm trying to think through the implications of deriving a time series of the variance of the original time series via moving windows. Having computed the derived series of SD values the next step that is commonly applied is to see if there is some trend in the derived series of SD values. As each value in the derived series depends to some extent on the previous values of the original series the values of the derived series are not independent. Thus a question that crops up frequently is how to account for that lack of independence.
Such computations (the moving windows) are often done to time series to look for evidence of indicators (increasing variance, increasing AR(1) coefficient) of impending threshold response (so-called critical transitions).