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

Taras Shevchenko National University, vul. Volodymyrska 64, Kyiv, 01601, Ukraine. yap@univ.kiev.ua

Received: 18 March 2009
Revised: 17 November 2009

Abstract

Let, for each t ∈ T, ψ(t, ۔) be a random measure on the Borel σ-algebra in ℝ^d such that Eψ(t, ℝ^d)^k < ∞ for all k and let (t, ۔) be its characteristic function. We call the function (t₁,…, t_l ; z₁,…, z_l) = of arguments l ∈ ℕ, t₁, t₂… ∈ T, z₁, z₂ ∈ ℝ^d 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 ψ_n(t, B) = µ{x : ξ_n(t, x) ∈ B}, where μ is a nonrandom finite measure and, for each n, ξ_n is a time-dependent random field.