## Ji\v r\'{i} Jan\'a\v cek

*Variance of periodic measure of bounded set with random position *

Comment.Math.Univ.Carolin. 47,3 (2006) 443-455. **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

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