Under fat tails, outliers drive movements in the mean
The law of large numbers fails when dealing with Fat Tails. Thus the influence of outliers is insufficiently diluted by their probably of occurrence. This means they move the mean when they occur. This is important, as many important economic variables aggregate fat tailed components (which is not always obvious, as in the case of inflation). The idiosyncratic fluctuations (The Granular Origins of Aggregate Fluctuations) of these components is often large enough that most of the period to period movement is driven by the outliers rather than the median component. The median could be flat over time, yet the weighted mean will bounce around. Thus, Volatility is information about the underlying components.
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Transclude of 2022-01-10#406215Weighted ACV