Variance of periodic measure of bounded set with random position. (English) Zbl 1150.62315

Summary: The principal term in the asymptotic expansion of the variance of the periodic measure of a ball in \(\mathbb R^d\) under uniform random shift is proportional to the \((d+1)\)st power of the grid scaling factor. This result remains valid for a bounded set in \(\mathbb R^d\) with sufficiently smooth isotropic covariogram under a uniform random shift and an isotropic rotation, and the asymptotic term is proportional also to the \((d-1)\)-dimensional measure of the object boundary. The related coefficients are calculated for various periodic grids constructed from affine sets.


62E20 Asymptotic distribution theory in statistics
60E99 Distribution theory
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