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Extremal behavior of recurrent random sequences

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Abstract

A process Y n , n ≥ 1, satisfying the stochastic recurrent equation Y n = A n Y n−1 + B n , n ≥ 1, Y 0 ≥ 0, is studied in the paper; here (A n , B n ), n ≥ 1, are independent identically distributed pairs of nonnegative random variables. The cases when the values A n have a lognormal and log-Laplace distributions are considered. The tail index κ (for a stationary distribution) and the extremal index ϑ are studied. In the lognormal case, κ is determined and some useful properties of ϑ are established. In the log-Laplace case the both characteristics are obtained in explicit form.

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References

  1. W. Vervaat, “On a Stochastic Difference Equation and a Representation of Nonnegative Infinitely Divisible Random Variables,” Adv. Appl. Probab. 11, 750 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  2. L. de Haan, S. I. Resnick, H. Rootzen, and G. C. de Vries, “Extremal Behavior of Solutions to a Stochastic Difference Equation with Applications to ARCH Processes,” Stochast. Process. and Appl. 32(1), 213 (1989).

    Article  MATH  Google Scholar 

  3. T. J. Kozubowski and K. Podgórski, “Log-Laplace Distributions,” Int. Math. J. 3(4), 467 (2003).

    MATH  MathSciNet  Google Scholar 

  4. P. Embrechts, C. P. Klüppelberg, and T. Mikosh, Modelling Extremal Events for Insurance and Finance (Springer, Berlin, 2003).

    Google Scholar 

  5. M. Fréchet, “Sur les Formules de Réepartition des Revenus,” Rev. Inst. Int. Statist. 7(1), 32 (1939).

    Article  Google Scholar 

  6. T. Inoue, On Income Distribution: the Welfare Implication of the General Equilibrium Model and the Stochastic. Processes of Income Distribution Formation, PhD Thesis (Univ. Minnesota, 1978).

  7. P. Embrechts, R. Frey, and A. J. McNeil, Quantitative Risk Management (Princeton Univ. Press, 2005).

  8. H. Kesten, “Random Difference Equations and Renewal Theory for Products of Random Matrices,” Acta Math. 131, 207 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  9. M. R. Leadbetter, G. Lindgren, and H. Rootzen, Extremes and Related Properties of Random Sequences and Processes (Springer, New York, 1983; Mir, Moscow, 1989).

    MATH  Google Scholar 

  10. W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1971; Mir, Moscow, 1967), Vol. 2.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia

    O. C. Novitskaya & E. B. Yatsalo

Authors
  1. O. C. Novitskaya
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  2. E. B. Yatsalo
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Additional information

Original Russian Text © O.C. Novitskaya and E.B. Yatsalo, 2008, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2008, Vol. 63, No. 5, pp. 6–10.

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Novitskaya, O.C., Yatsalo, E.B. Extremal behavior of recurrent random sequences. Moscow Univ. Math. Bull. 63, 179–182 (2008). https://doi.org/10.3103/S0027132208050021

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  • DOI: https://doi.org/10.3103/S0027132208050021

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