Asymptotic study of measure-valued processes related to stochastic geometry (Q2739927)

Let \((Q,{\mathcal G},\mu)\) be a measurable space, let \(\mu\) be a finite measure, and let \((\alpha_{n})\) be a sequence of \(R^{d}\)-valued measurable random functions of \(t\in R_{+}, q\in Q\). Let us define the measure-valued processes \((\Psi_{n})\) by \(\Psi_{n}(t)=\psi_{n}(t,\cdot)\), \(\psi_{n}(t,B)=\mu\{q:\alpha_{n}(t,q)\in B\}\), \(B\in{\mathcal B}^{d}\). The goal of this paper is to state in terms of \(\alpha_{n}\) conditions for the weak convergence (in some sense) of \((\Psi_{n})\) and to specify the limiting process. A geometric illustration is presented.