Geodesic
Variance of periodic measure of bounded set with random position
Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 443-455.

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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.
Classification : 60E99, 62D05, 62E20, 62J10
Keywords: periodic measure; variance 
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     title = {Variance of periodic measure of bounded set with random position},
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     zbl = {1150.62315},
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Janáček, Jiří. Variance of periodic measure of bounded set with random position. Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 443-455. http://geodesic.mathdoc.fr/item/CMUC_2006__47_3_a6/