I have a time series $X_t \sim N(0, 1)$. There is a single outlier at index 347, at 8.5 standard deviations from the mean. If I now compute a rolling window standard deviation of $X_t$ with window size 100, we of course see a large spike at index 347.
Once this observation has fallen out of the 100 period window, the rolling standard deviation falls back down to normal levels at index 447 (347+100=447). This is all as expected.
My question is, what is this phenomenon formally called, because I wan't to research it more to be able to understand it and mitigate it. Specifically, once the outlier observation falls out of the rolling window, I do not want the standard deviation estimation to jump back down to normal levels.
I know some ways to mitigate this may be to time weigh or exponential-decay weigh the observations prior to estimating the standard deviation. But I am wondering what this phenomenon is called, if it even has a name, and if there are potentially other ways to mitigate it.
EDIT: Just for clarity, I am using an equal weighted rolling window, and at each point in time $t$ I can only use observations up to and including $t$ i.e. at time $t$, all observations $x$ used to estimate the standard deviation must be in $x_l, l\leq t$. This is to avoid lookahead bias. So any method that uses "future" information is not allowed.
