Geometrical detection of weak non-Gaussianity upon coarse-graining

Measures of the non-Gaussianity of a random field depend on how accurately one is able to measure the field. If a signal measured at a certain point is to be averaged with its surroundings, or coarse-grained, the magnitude of its non-Gaussian component can vary. In this article, we investigate the variation of the "apparent" non-Gaussianity, as a function of the coarse-graining length, when we measure non-Gaussianity using the statistics of extrema in the field. We derive how the relative difference between maxima and minima -- which is a geometrical measure of the field's non-Gaussianity -- behaves as the field is coarse-grained over increasingly larger length scales. Measuring this function can give extra information about the non-Gaussian statistics and facilitate its detection.