Gaussian and Non-Gaussian Stable Limit Laws in Wicksell's Corpuscle Problem

Suppose that a homogeneous system of spherical particles (d-spheres) with independent identically distributed radii is contained in some opaque d-dimensional body, and one is interested to estimate the common radius distribution. The only information one can get is by making a cross-section of that body with an s-flat (0 <= s <= d-1) and measuring the radii of the s-spheres and heir midpoints. The analytical solution of (the hyper-stereological version of) Wicksell's corpuscle problem is used to construct an empirical radius distribution of the d-spheres. In this paper we study the asymptotic behaviour of this empirical radius distribution for s = d-1 and s = d-2 under the assumption that the intersection volume becomes unboundedly large and the point process of the midpoints of the d-spheres is Brillinger-mixing. Among others we generalize and extend some results obtained in [1] and [2] under the Poisson assumption for the special case d=3 and s=2