Comparisons and asymptotics for empty space hazard functions of germ-grain models

TL;DR

This paper analyzes the stochastic properties and asymptotic behavior of the empty space hazard function in stationary germ-grain models, highlighting how cluster variability and spread influence empty space characteristics.

Contribution

It introduces new insights into the empty space hazard function for germ-grain models, including effects of cluster size variability and spatial spread, with asymptotic analysis.

Findings

Greater cluster size variability increases empty space hazard.

More spread-out clusters lead to higher empty space hazard.

Asymptotic behaviors are characterized at zero and infinity.

Abstract

We study stochastic properties of the empty space for stationary germ-grain models in $\R^d$, in particular we deal with the inner radius of the empty space with respect to a general structuring element which is allowed to be lower-dimensional. We consider Poisson cluster germ-grain models and Boolean models with grains that are clusters of convex bodies and show that more variable size of clusters results in stochastically greater empty space in terms of the empty space hazard function. We also study impact of clusters being more spread in the space on the value of the empty space hazard. Further we obtain asymptotic behavior of the empty space hazard functions at zero and at infinity.