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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.

##### MSC:
 6.2e+21 Asymptotic distribution theory in statistics 6e+100 Distribution theory
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