Multiple-point hit distribution functions and vague convergence of related measures

For a stationary and isotropic random closed set Z in R it is a well-known fact that its covariance C(t) and its spherical contact distribution function H̃s(t) admit at t = 0 a derivative which is a multiple of the surface intensity of Z. Within the quite general setting of gentle sets, Kiderlen and Rataj [10] show a more general result (covering both previous cases) for the derivative of a hit distribution function of Z with respect to a structuring element which only needs to be compact and should contain the origin. Using this general setting we introduce m-point hit distribution functions of Z, m ≥ 2, and show how they are related to the mth-order surface product density of Z. This also generalizes a result of Ballani [1] for the two-point spherical contact distribution function of a germ-grain model.