Limit theorems for measure-valued processes of the level-exceedance type
ESAIM: Probability and Statistics, Tome 15 (2011), pp. 291-319.

Let, for each tT, 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
Mots clés : measure-valued process, covaristic, convergence, relative compactness
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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/

[1] V. Beneš and J. Rataj, Stochastic Geometry: Selected Topics. Kluwer, Dordrecht (2004). | MR | Zbl

[2] D.J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Elementary Theory and Methods. Springer, New York (2002) Vol. 1. | MR | Zbl

[3] D. Dawson, Measure-Valued Markov Processes. Lect. Notes Math. 1541 (1991). | Zbl

[4] I.I. Gikhman and A.V. Skorokhod, Stochastic Differential Equations and Their Applications. Naukova Dumka, Kiev (1982) (Russian). | MR | Zbl

[5] P. Hall, Introduction to the Theory of Coverage Processes. Wiley, New York (1988). | MR | Zbl

[6] P.J. Huber, Robust Statistics. Wiley, New York (1981). | MR | Zbl

[7] J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes. Springer, Berlin (1987). | MR | Zbl

[8] O. Kallenberg, Random Measures. Academic Press, New York, London; Akademie-Verlag, Berlin (1988). | Zbl

[9] A.N. Kolmogorov, On the statistical theory of metal crystallization. Izvestiya Akademii Nauk SSSR [Bull. Acad. Sci. USSR] (1937), Issue 3, 355-359 (Russian) [ English translation in: Selected Works of A.N. Kolmogorov, Probability Theory and Mathematical Statistics. Springer, New York (1992), Vol. 2, 188-192.

[10] D.L. Mcleish, An extended martingale principle. Ann. Prob. 6 (1978) 144-150. | MR | Zbl

[11] Yu. V. Prokhorov, Convergence of random processes and limit theorems in probability theory. Th. Prob. Appl. 1 (1956) 157-214. | Zbl

[12] A.N. Shiryaev, Probability. Springer, Berlin (1996)

[13] A.V. Skorokhod, Limit theorems for stochastic processes. Th. Prob. Appl. 1 (1956) 261-290. | Zbl

[14] A.V. Skorokhod, Studies in the Theory of Random Processes. McGraw-Hill, New York (1965). | MR | Zbl

[15] A.V. Skorokhod, Stochastic Equations for Complex Systems. Kluwer, Dordrecht (1987). | MR | Zbl

[16] D. Stoyan, W.S. Kendall and J. Mecke, Stochastic Geometry and Its Applications. Akademie-Verlag, Berlin (1987). | MR | Zbl

[17] N.N. Vakhania, V.I. Tarieladze and S.A Chobanian, Probability Distributions in Banach Spaces. Reidel Pub. Co., Dordrecht-Boston (1987). | Zbl

[18] A.P. Yurachkivsky, Covariance-characteristic functions of random measures and their applications to stochastic geometry. Dopovidi Natsionalnoĭi Akademii Nauk Ukrainy (1999), Issue 5, 49-54. | MR | Zbl

[19] A.P. Yurachkivsky, Some applications of stochastic analysis to stochastic geometry. Th. Stoch. Proc. 5 (1999) 242-257. | MR | Zbl

[20] A.P. Yurachkivsky, Covaristic functions of random measures and their applications. Th. Prob. Math. Stat. 60 (2000) 187-197. | Zbl

[21] A.P. Yurachkivsky, A generalization of a problem of stochastic geometry and related measure-valued processes. Ukr. Math. J. 52 (2000) 600-613. | Zbl

[22] A.P. Yurachkivsky, Two deterministic functional characteristics of a random measure. Th. Prob. Math. Stat. 65 (2002) 189-197. | MR | Zbl

[23] A.P. Yurachkivsky, Asymptotic study of measure-valued processes generated by randomly moving particles. Random Operators Stoch. Equations 10 (2002) 233-252. | MR | Zbl

[24] A. Yurachkivsky, A criterion for relative compactness of a sequence of measure-valued random processes. Acta Appl. Math. 79 (2003) 157-164. | MR | Zbl

[25] A.P. Yurachkivsky and G.G. Shapovalov, On the kinetics of amorphization under ion implantation, in: Frontiers in Nanoscale Science of Micron/Submicron Devices, NATO ASI, Series E: Applied Sciences, edited by A.-P. Jauho and E.V. Buzaneva. Kluwer, Dordrecht (1996) Vol. 328, 413-416.

[26] H. Zessin, The method of moments for random measures. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 62 (1983) 359-409. | MR | Zbl

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