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

Yurachkivsky, Andriy

ESAIM: Probability and Statistics, Tome 15 (2011), pp. 291-319.

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 $\widehat{\psi }$(t, ۔) be its characteristic function. We call the function $\widehat{\psi }$ (t₁,…, t_l ; z₁,…, z_l) = $𝖤{\prod }_{j=1}^l\widehat{\psi }(t_j,z_j)$ 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.

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