Limit theorems for measure-valued processes of the level-exceedance type

Abstract

Let, for each t ∈ T , ψ (t , ۔) be a random measure on the Borel σ -algebra in ℝd such that Eψ (t , ℝd )k and let (t , ۔) be its characteristic function. We call the function (t 1 ,…, t l ; z 1 ,…, z l ) = of arguments l ∈ ℕ, t 1 , t 2 … ∈ T , z 1 , z 2 ∈ ℝd the covaristic of the measure-valued random function (MVRF) ψ (۔, ۔). A general limit theorem for MVRF's in terms of covaristics is proved and applied to functions of the kind ψn (t, B ) = µ {x : ξn (t, x ) ∈ B }, where μ is a nonrandom finite measure and, for each n, ξn is a time-dependent random field.