Variance of periodic measure of bounded set with random position. (English) Zbl 1150.62315

Janáček, Jiří

Commentat. Math. Univ. Carol. 47, No. 3, 443-455 (2006).

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:

62E20	Asymptotic distribution theory in statistics
60E99	Distribution theory