Normal convergence of nonlocalised geometric functionals and shot-noise excursions

Abstract

This article presents a complete second order theory for a large class of geometric functionals on homogeneous Poisson input. In particular, the results don't require the existence of a radius of stabilisation. Hence they can be applied to geometric functionals of spatial shot-noise fields excursions such as volume, perimeter, or Euler characteristic (the method still applies to stabilising functionals). More generally, it must be checked that a local contribution to the functional is not strongly affected under a perturbation of the input far away. In this case the exact asymptotic variance is given, as well as the likely optimal speed of convergence in the central limit theorem. This goes through a general mixing-type condition that adapts nicely to both proving asymptotic normality and that variance is of volume order.

Intrinsic volumes of sublevel sets

“…Sublevel sets are also studied in probability theory, where superlevel sets of random fields are called excursion sets; see for instance the books [1,4] and the articles [2,17,23]. The importance of the formulas obtained in this paper for the study of random fields (as studied in [2]), compared to existing Kac-Rice formulas, stems from the fact that they are in closed form as opposed to being limits of integrals depending on a parameter.…”

Percolation of the excursion sets of planar symmetric shot noise fields

“…In the analysis of telecommunications networks, the percolative properties of {f + ℓ ≥ 0} are of high importance in analysing the connectivity of the network [BB15]. More generally, the geometric properties of {f + ℓ ≥ 0} have been the focus of many other studies [BD20, LR19,BST12].…”

Testing marginal symmetry of digital noise images through the perimeter of excursion sets

“…Moreover, Central Limit Theorems (CLT) have been proven for Gaussian fields [16,23] that paved the road for statistical hypothesis tests. Several extensions for non-Gaussian fields, namely shot noise random fields are also available for means of LK curvatures [24] and associated CLT's [25]. However the framework of smooth fields, indexed by continuous space variables, is not appropriate with the discrete framework induced by digital images.…”