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

ESAIM: Probability and Statistics (2012)

- Volume: 15, page 291-319
- ISSN: 1292-8100

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topYurachkivsky, Andriy. "Limit theorems for measure-valued processes of the level-exceedance type." ESAIM: Probability and Statistics 15 (2012): 291-319. <http://eudml.org/doc/222469>.

@article{Yurachkivsky2012,

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.
},

author = {Yurachkivsky, Andriy},

journal = {ESAIM: Probability and Statistics},

keywords = {Measure-valued process; covaristic; convergence; relative compactness; measure-valued process; random fields},

language = {eng},

month = {1},

pages = {291-319},

publisher = {EDP Sciences},

title = {Limit theorems for measure-valued processes of the level-exceedance type},

url = {http://eudml.org/doc/222469},

volume = {15},

year = {2012},

}

TY - JOUR

AU - Yurachkivsky, Andriy

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

JO - ESAIM: Probability and Statistics

DA - 2012/1//

PB - EDP Sciences

VL - 15

SP - 291

EP - 319

AB -
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.

LA - eng

KW - Measure-valued process; covaristic; convergence; relative compactness; measure-valued process; random fields

UR - http://eudml.org/doc/222469

ER -

## References

top- 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. Zbl1026.60061
- D. Dawson, Measure-Valued Markov Processes. Lect. Notes Math.1541 (1991). Zbl0799.60080
- I.I. Gikhman and A.V. Skorokhod, Stochastic Differential Equations and Their Applications. Naukova Dumka, Kiev (1982) (Russian). Zbl0557.60041
- P. Hall, Introduction to the Theory of Coverage Processes. Wiley, New York (1988). Zbl0659.60024
- P.J. Huber, Robust Statistics. Wiley, New York (1981). Zbl0536.62025
- J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes. Springer, Berlin (1987). Zbl0635.60021
- 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.
- D.L. McLeish, An extended martingale principle. Ann. Prob.6 (1978) 144–150. Zbl0379.60046
- Yu. V. Prokhorov, Convergence of random processes and limit theorems in probability theory. Th. Prob. Appl.1 (1956) 157–214.
- A.N. Shiryaev, Probability. Springer, Berlin (1996)
- A.V. Skorokhod, Limit theorems for stochastic processes. Th. Prob. Appl.1 (1956) 261–290. Zbl0074.33802
- A.V. Skorokhod, Studies in the Theory of Random Processes. McGraw–Hill, New York (1965). Zbl0146.37701
- 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). Zbl0622.60019
- 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.
- A.P. Yurachkivsky, Some applications of stochastic analysis to stochastic geometry. Th. Stoch. Proc.5 (1999) 242–257. Zbl0993.60045
- A.P. Yurachkivsky, Covaristic functions of random measures and their applications. Th. Prob. Math. Stat.60 (2000) 187–197.
- A.P. Yurachkivsky, A generalization of a problem of stochastic geometry and related measure-valued processes. Ukr. Math. J.52 (2000) 600–613.
- A.P. Yurachkivsky, Two deterministic functional characteristics of a random measure. Th. Prob. Math. Stat.65 (2002) 189–197.
- A.P. Yurachkivsky, Asymptotic study of measure-valued processes generated by randomly moving particles. Random Operators Stoch. Equations10 (2002) 233–252. Zbl1004.60049
- A. Yurachkivsky, A criterion for relative compactness of a sequence of measure-valued random processes. Acta Appl. Math.79 (2003) 157–164. Zbl1030.60037
- 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.
- H. Zessin, The method of moments for random measures. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete62 (1983) 359–409. Zbl0489.60063

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