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

DOI : 10.1051/ps/2010004
Classification : 60G57, 60F17
Keywords: 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},
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     url = {https://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

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