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 (t, ۔) be its characteristic function. We call the function (t1,…, tl ; z1,…, zl) = 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.
Mots clés : measure-valued process, covaristic, convergence, relative compactness
@article{PS_2011__15__291_0, 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/} }
TY - JOUR AU - Yurachkivsky, Andriy TI - Limit theorems for measure-valued processes of the level-exceedance type JO - ESAIM: Probability and Statistics PY - 2011 SP - 291 EP - 319 VL - 15 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2010004/ DO - 10.1051/ps/2010004 LA - en ID - PS_2011__15__291_0 ER -
%0 Journal Article %A Yurachkivsky, Andriy %T Limit theorems for measure-valued processes of the level-exceedance type %J ESAIM: Probability and Statistics %D 2011 %P 291-319 %V 15 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2010004/ %R 10.1051/ps/2010004 %G en %F PS_2011__15__291_0
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] Stochastic Geometry: Selected Topics. Kluwer, Dordrecht (2004). | MR | Zbl
and ,[2] An Introduction to the Theory of Point Processes. Elementary Theory and Methods. Springer, New York (2002) Vol. 1. | MR | Zbl
and ,[3] Measure-Valued Markov Processes. Lect. Notes Math. 1541 (1991). | Zbl
,[4] Stochastic Differential Equations and Their Applications. Naukova Dumka, Kiev (1982) (Russian). | MR | Zbl
and ,[5] Introduction to the Theory of Coverage Processes. Wiley, New York (1988). | MR | Zbl
,[6] Robust Statistics. Wiley, New York (1981). | MR | Zbl
,[7] Limit Theorems for Stochastic Processes. Springer, Berlin (1987). | MR | Zbl
and ,[8] Random Measures. Academic Press, New York, London; Akademie-Verlag, Berlin (1988). | Zbl
,[9] 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] An extended martingale principle. Ann. Prob. 6 (1978) 144-150. | MR | Zbl
,[11] Convergence of random processes and limit theorems in probability theory. Th. Prob. Appl. 1 (1956) 157-214. | Zbl
,[12] Probability. Springer, Berlin (1996)
,[13] Limit theorems for stochastic processes. Th. Prob. Appl. 1 (1956) 261-290. | Zbl
,[14] Studies in the Theory of Random Processes. McGraw-Hill, New York (1965). | MR | Zbl
,[15] Stochastic Equations for Complex Systems. Kluwer, Dordrecht (1987). | MR | Zbl
,[16] Stochastic Geometry and Its Applications. Akademie-Verlag, Berlin (1987). | MR | Zbl
, and ,[17] Probability Distributions in Banach Spaces. Reidel Pub. Co., Dordrecht-Boston (1987). | Zbl
, and ,[18] 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] Some applications of stochastic analysis to stochastic geometry. Th. Stoch. Proc. 5 (1999) 242-257. | MR | Zbl
,[20] Covaristic functions of random measures and their applications. Th. Prob. Math. Stat. 60 (2000) 187-197. | Zbl
,[21] A generalization of a problem of stochastic geometry and related measure-valued processes. Ukr. Math. J. 52 (2000) 600-613. | Zbl
,[22] Two deterministic functional characteristics of a random measure. Th. Prob. Math. Stat. 65 (2002) 189-197. | MR | Zbl
,[23] Asymptotic study of measure-valued processes generated by randomly moving particles. Random Operators Stoch. Equations 10 (2002) 233-252. | MR | Zbl
,[24] A criterion for relative compactness of a sequence of measure-valued random processes. Acta Appl. Math. 79 (2003) 157-164. | MR | Zbl
,[25] 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.
and ,[26] The method of moments for random measures. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 62 (1983) 359-409. | MR | Zbl
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