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

Andriy Yurachkivsky

ESAIM: Probability and Statistics (2011)

  • Volume: 15, page 291-319
  • ISSN: 1292-8100

Abstract

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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 ψ ^ (t, ۔) be its characteristic function. We call the function ψ ^ (t1,…, tl ; z1,…, zl) = 𝖤 ∏ j = 1 l ψ ^ ( t j , z j ) 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 and, for each n, ξn is a time-dependent random field.

How to cite

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Yurachkivsky, Andriy. "Limit theorems for measure-valued processes of the level-exceedance type." ESAIM: Probability and Statistics 15 (2011): 291-319. <http://eudml.org/doc/277148>.

@article{Yurachkivsky2011,
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 &lt; ∞ for all kand let $\widehat\{\psi \}$(t, ۔) be its characteristic function. We call the function $\widehat\{\psi \}$ (t1,…, tl ; z1,…, zl) = $\{\sf E\}\prod ^l_\{j=1\}\widehat\{\psi \}(t_j, z_j)$ 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 and, for each n, ξn is a time-dependent random field.},
author = {Yurachkivsky, Andriy},
journal = {ESAIM: Probability and Statistics},
keywords = {measure-valued process; covaristic; convergence; relative compactness; random fields},
language = {eng},
pages = {291-319},
publisher = {EDP-Sciences},
title = {Limit theorems for measure-valued processes of the level-exceedance type},
url = {http://eudml.org/doc/277148},
volume = {15},
year = {2011},
}

TY - JOUR
AU - Yurachkivsky, Andriy
TI - Limit theorems for measure-valued processes of the level-exceedance type
JO - ESAIM: Probability and Statistics
PY - 2011
PB - EDP-Sciences
VL - 15
SP - 291
EP - 319
AB - Let, for each t ∈ T, ψ(t, ۔) be a random measure on the Borel σ-algebra in ℝd such that Eψ(t, ℝd)k &lt; ∞ for all kand let $\widehat{\psi }$(t, ۔) be its characteristic function. We call the function $\widehat{\psi }$ (t1,…, tl ; z1,…, zl) = ${\sf E}\prod ^l_{j=1}\widehat{\psi }(t_j, z_j)$ 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 and, for each n, ξn is a time-dependent random field.
LA - eng
KW - measure-valued process; covaristic; convergence; relative compactness; random fields
UR - http://eudml.org/doc/277148
ER -

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