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

Andriy Yurachkivsky

ESAIM: Probability and Statistics (2012)

  • 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 k and 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 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 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 (2012): 291-319. <http://eudml.org/doc/222469>.

@article{Yurachkivsky2012,
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 < ∞ for all k and 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 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 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; measure-valued process; random fields},
language = {eng},
month = {1},
pages = {291-319},
publisher = {EDP Sciences},
title = {Limit theorems for measure-valued processes of the level-exceedance type},
url = {http://eudml.org/doc/222469},
volume = {15},
year = {2012},
}

TY - JOUR
AU - Yurachkivsky, Andriy
TI - Limit theorems for measure-valued processes of the level-exceedance type
JO - ESAIM: Probability and Statistics
DA - 2012/1//
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 < ∞ for all k and 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 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 and, for each n, ξn is a time-dependent random field.

LA - eng
KW - Measure-valued process; covaristic; convergence; relative compactness; measure-valued process; random fields
UR - http://eudml.org/doc/222469
ER -

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