Mean densities and spherical contact distribution function of inhomogeneous Boolean models

The mean density of a random closed set Θ in d with Hausdorff dimension n is the Radon-Nikodym derivative of the expected measure [Hn(Θ ∩·)] induced by Θ with respect to the usual d-dimensional Lebesgue measure. Starting from an open problem posed by Matheron in [24, pp. 50-51], we consider here inhomogeneous Boolean models Ξ in d with integer Hausdorff dimension n ∈ 0,…, d, and we study the mean density of their boundary (which is their mean density if n < d) and the differentiability of their spherical contact distribution function HΞ, under general regularity assumptions on the typical grain, related to the existence of its (outer) Minkowski content. In particular, we provide an explicit formula for ∂2HΞ(r, x)/(∂r2) at r = 0 for a class of Boolean models, whose typical grain has positive reach; known results for stationary Boolean models with convex grains follows then as a particular case. Examples and statistical applications are also discussed.