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

Yurachkivsky, Andriy

doi:10.1051/ps/2010004

Yurachkivsky, Andriy

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

Résumé

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 $\hat{ψ}$ (t, ۔) be its characteristic function. We call the function $\hat{ψ}$ (t₁,…, t_l ; z₁,…, z_l) = $𝖤 \prod_{j = 1}^{l} \hat{ψ} (t_{j}, z_{j})$ 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.

MR Zbl

DOI : 10.1051/ps/2010004

Classification : 60G57, 60F17
Mots clés : measure-valued process, covaristic, convergence, relative compactness

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     author = {Yurachkivsky, Andriy},
     title = {Limit theorems for measure-valued processes of the level-exceedance type},
     journal = {ESAIM: Probability and Statistics},
     pages = {291--319},
     publisher = {EDP-Sciences},
     volume = {15},
     year = {2011},
     doi = {10.1051/ps/2010004},
     mrnumber = {2870517},
     zbl = {1296.60131},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2010004/}
}

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Yurachkivsky, Andriy. Limit theorems for measure-valued processes of the level-exceedance type. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 291-319. doi : 10.1051/ps/2010004. http://www.numdam.org/articles/10.1051/ps/2010004/

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