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Limit theorems for measure-valued processes of the level-exceedance type
Published online by Cambridge University Press: 05 January 2012
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.
- Type
- Research Article
- Information
- ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 291 - 319
- Copyright
- © EDP Sciences, SMAI, 2011
References
V. Beneš and J. Rataj, Stochastic Geometry: Selected Topics. Kluwer, Dordrecht (2004).
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.
D. Dawson, Measure-Valued Markov Processes. Lect. Notes Math. 1541 (1991).
I.I. Gikhman and A.V. Skorokhod, Stochastic Differential Equations and Their Applications. Naukova Dumka, Kiev (1982) (Russian).
P. Hall, Introduction to the Theory of Coverage Processes. Wiley, New York (1988).
P.J. Huber, Robust Statistics. Wiley, New York (1981).
J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes. Springer, Berlin (1987).
O. Kallenberg, Random Measures. Academic Press, New York, London; Akademie-Verlag, Berlin (1988).
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.
Prokhorov, Yu. V., Convergence of random processes and limit theorems in probability theory. Th. Prob. Appl. 1 (1956) 157–214.
CrossRef
A.N. Shiryaev, Probability. Springer, Berlin (1996)
Skorokhod, A.V., Limit theorems for stochastic processes. Th. Prob. Appl. 1 (1956) 261–290.
CrossRef
A.V. Skorokhod, Studies in the Theory of Random Processes. McGraw–Hill, New York (1965).
A.V. Skorokhod, Stochastic Equations for Complex Systems. Kluwer, Dordrecht (1987).
D. Stoyan, W.S. Kendall and J. Mecke, Stochastic Geometry and Its Applications. Akademie-Verlag, Berlin (1987).
N.N. Vakhania, V.I. Tarieladze and S.A Chobanian, Probability Distributions in Banach Spaces. Reidel Pub. Co., Dordrecht-Boston (1987).
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.
Yurachkivsky, A.P., Some applications of stochastic analysis to stochastic geometry. Th. Stoch. Proc. 5 (1999) 242–257.
Yurachkivsky, A.P., Covaristic functions of random measures and their applications. Th. Prob. Math. Stat. 60 (2000) 187–197.
Yurachkivsky, A.P., A generalization of a problem of stochastic geometry and related measure-valued processes. Ukr. Math. J. 52 (2000) 600–613.
CrossRef
Yurachkivsky, A.P., Two deterministic functional characteristics of a random measure. Th. Prob. Math. Stat. 65 (2002) 189–197.
Yurachkivsky, A.P., Asymptotic study of measure-valued processes generated by randomly moving particles. Random Operators Stoch. Equations 10 (2002) 233–252.
Yurachkivsky, A., A criterion for relative compactness of a sequence of measure-valued random processes. Acta Appl. Math. 79 (2003) 157–164.
CrossRef
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.
Zessin, H., The method of moments for random measures. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 62 (1983) 359–409.
CrossRef