Level crossings and other level functionals of stationary Gaussian processes.
Level sets of multiparameter Brownian motions.
Level sets of random fields and applications: specular points and wave crests.
Limit Theorem for High Level -upcrossings by -field
Limit theorems for measure-valued processes of the level-exceedance type
Let, for each t∈T, ψ(t, ۔) be a random measure on the Borel σ-algebra in ℝd such that Eψ(t, ℝd)k < ∞ for all k and let (t, ۔) be its characteristic function. We call the function (t1,…, tl ; z1,…, zl) = of arguments l∈ ℕ, t1, t2… ∈T, z1, z2∈ 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...
Limit theorems for measure-valued processes of the level-exceedance type
Let, for each t ∈ T, ψ(t, ۔) be a random measure on the Borel σ-algebra in ℝd such that Eψ(t, ℝd)k < ∞ for all kand let (t, ۔) be its characteristic function. We call the function (t1,…, tl ; z1,…, zl) = of argumentsl ∈ ℕ, t1, t2… ∈ T, z1, z2 ∈ 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...