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Siberian Mathematical Journal

, Volume 60, Issue 3, pp 516–525 | Cite as

The Discrete Wiener-Hopf Equation with Probability Kernel of Oscillating Type

  • M. S. SgibnevEmail author
Article
  • 11 Downloads

Abstract

We prove the existence of a solution to the discrete inhomogeneous Wiener-Hopf equation whose kernel is an arithmetic probability distribution generating an oscillating random walk. Asymptotic properties of the solution are established depending on the properties of the inhomogeneous term of the equation and its kernel.

Keywords

discrete Wiener-Hopf equation inhomogeneous equation asymptotic behavior arithmetic distribution oscillating type 

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References

  1. 1.
    Krein M. G., “Integral equations on the half-line with kernel depending on the difference of the arguments,” Uspekhi Mat. Nauk, vol. 13, no. 5, 3–120 (1958).MathSciNetGoogle Scholar
  2. 2.
    Feller W., An Introduction to Probability Theory and Its Applications. vol. 2, John Wiley and Sons, Inc., New York etc. (1966).Google Scholar
  3. 3.
    Kolmogorov A. N. and Fomin S. V., Elements of the Theory of Functions and Functional Analysis, Dover Publications, Mineola (1999).Google Scholar
  4. 4.
    Dmitriev V. I., “The Wiener-Hopf equation,” n: Mathematical Encyclopedia [Russian], Sovetskaya Èntsiklopediya, Moscow, 1977, vol. 1, 697–698.Google Scholar
  5. 5.
    Gakhov F. D. and Cherskii Yu. I., Equations of Convolution Type [Russian], Nauka, Moscow (1978).zbMATHGoogle Scholar
  6. 6.
    Halmos P. R., Measure Theory, Springer-Verlag, New York, Heidelberg, and Berlin (1974).zbMATHGoogle Scholar
  7. 7.
    Sgibnev M. S., “Homogeneous conservative Wiener-Hopf equation,” Sb. Math., vol. 198, no. 9, 1341–1350 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Doney R. A., “Moments of ladder heights in random walks,” J. Appl. Probab., vol. 17, 248–252 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Arabadzhyan L. G., “Discrete Wiener-Hopf equations in the conservative case,” in: Mathematical Analysis and Its Applications, Erevan, 1980, 26–36.Google Scholar
  10. 10.
    Arabadzhyan L. G., “The conservative Wiener-Hopf equation,” Izv. Nats. Akad. Nauk Armenii Mat., vol. 16, no. 1, 65–80 (1981).MathSciNetzbMATHGoogle Scholar
  11. 11.
    Arabadzhyan L. G. and Engibaryan N. B., “Convolution equations and nonlinear functional equations,” in: Mathematical Analysis. Vol. 22 [Russian], VINITI, Moscow, 1984, 175–245 (Itogi Nauki i Tekhniki).MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

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