Abstract:The principal term in the asymptotic expansion of the variance of the periodic measure of a ball in $\Bbb R^d$ 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 $\Bbb R^d$ 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.
Keywords: periodic measure, variance
AMS Subject Classification: 62J10, 62D05