Skip to main content
SpringerLink
Log in

Generalization of one problem of stochastic geometry and related measure-valued processes

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We prove a functional limit theorem for the measure of a domain in which the values of a time-dependent random field do not exceed a given level. We illustrate this theorem by a geometric model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. P. Hall, Introduction to the Theory of Coverage Processes, Wiley, New York 1988.

    MATH  Google Scholar 

  2. A. P. Yurachkivs’kyi, “The law of large numbers for the measure of a domain covered by a flow of random sets,” Teor. Ver. Mat. Statist., No. 55, 173–177 (1996).

    Google Scholar 

  3. A. P. Yurachkivs’kyi, “Limit theorems for the measures of sections of random sets,” Teor. Ver. Mat. Statist., No. 56, 177–182 (1997).

  4. A. P. Yurachkivs’kyi, “Phenomena of averaging for unions of random sets,” Dopov. Akad. Nauk Ukr., No. 6, 45–9 (1997).

    Google Scholar 

  5. A. P. Yurachkivsky, “Functional limit theorems for coverage processes,” Theory Stochast. Proc., 3(19), Nos. 3-4, 475–84 (1997).

    Google Scholar 

  6. A. P. Yurachkivs’kyi, “A functional limit theorem of the type of the law of large numbers for random reliefs,” Ukr. Mat. Zh., 51, 4, 542–552 (1999).

    Article  Google Scholar 

  7. A. N. Kolmogorov, “On the statistical theory of crystallization of metals,” Izv. Akad. Nauk SSSR, No. 3, 355–359 (1937).

    Google Scholar 

  8. W. A. Johnson and R. F. Mehl, “Reaction kinetics in processes of nucleation and growth,” Trans. Amer. Inst. Mining Met. Eng. Iron Steel, 135, 416–458 (1939).

    Google Scholar 

  9. V. Z. Belen’kii, Geometric Probability Models of Crystallization [in Russian] Nauka, Moscow 1980.

    Google Scholar 

  10. A. P. Yurachkivsky and G. G. Shapovalov, “On the kinetics of amorphization under ion implantation,” in: A.-P. Yauho and E. V. Buzaneva (eds.), Proc. NATO ASI “Frontiers in Nanoscale Sciences of Micron/Submicron Devices,” Kluwer/North Holland (1996), pp. 413–416.

    Google Scholar 

  11. J. Jacod and A. N. Shiryaev, Limit Theorems for Random Processes [in Russian], Vol. 1, Fizmatlit, Moscow (1994).

    Google Scholar 

  12. D. L. McLeish, “An extended martingale principle,” Ann. Probab., 6, 144–150 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  13. V. V. Petrov, Sums of Independent Random Variables [in Russian] Nauka, Moscow 1972.

    Google Scholar 

  14. A. V. Skorokhod, “Limit theorems for random processes,” Teor. Ver. Primen., 1, No. 3, 289–319 (1956).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kiev University, kiev

    A. P. Yurachkivs’kyi

Authors
  1. A. P. Yurachkivs’kyi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yurachkivs’kyi, A.P. Generalization of one problem of stochastic geometry and related measure-valued processes. Ukr Math J 52, 600–613 (2000). https://doi.org/10.1007/BF02515399

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02515399

Keywords

Search

Navigation