Geodesic
Variance of periodic measure of bounded set with random position
Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 443-455 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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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     author = {Jan\'a\v{c}ek, Ji\v{r}{\'\i}},
     title = {Variance of periodic measure of bounded set with random position},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {443--455},
     year = {2006},
     volume = {47},
     number = {3},
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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/