Asymptotic results for multivariate estimators of the mean density of random closed sets

Abstract

The problem of the evaluation and estimation of the mean density of random closed sets in $\mathbb{R} ^{d}$ with integer Hausdorff dimension $0<n<d$, is of great interest in many different scientific and technological fields. Among the estimators of the mean density available in literature, the so-called “Minkowski content”-based estimator reveals its benefits in applications in the non-stationary cases. We introduce here a multivariate version of such estimator, and we study its asymptotical properties by means of large and moderate deviation results. In particular we prove that the estimator is strongly consistent and asymptotically Normal. Furthermore we also provide confidence regions for the mean density of the involved random closed set in $m\geq1$ distinct points $x_{1},\ldots,x_{m}\in\mathbb{R} ^{d}$.