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Random Sums of Random Variables and Vectors

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  • Omey, Edward

    ()
    (Hogeschool-Universiteit Brussel (HUB), Belgium
    Katholieke Universiteit Leuven, Belgium)

  • Vesilo, R.

    ()
    (Macquarie University, Sydney, Australia)

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    Abstract

    Let fX;Xi; i = 1; 2; :::g denote independent positive random variables having a common distribution function F(x) and, independent of X, let N denote an integer valued random variable. Using S(0) = 0 and S(n) = S(n ?? 1) + Xn, the random sum S(N) has distribution function G(x) = 1Xi=0 P(N = i)P(S(i) _ x) and tail distribution G(x) = 1 ?? G(x). In which case, we say that the distribution function G is subordinated to F with subordinator N. Under suitable conditions, it can be proved that G(x) s E(N)F(x) as x ! 1. In this paper, we extend some of the existing results. In the place of i.i.d. random variables, we use variables that are independent or variables that are asymptotically in- dependent. We also consider multivariate subordinated distribution functions.

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    File URL: https://lirias.hubrussel.be/bitstream/123456789/2469/1/09HRP09.pdf
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    Bibliographic Info

    Paper provided by Hogeschool-Universiteit Brussel, Faculteit Economie en Management in its series Working Papers with number 2009/09.

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    Length: 31 pages
    Date of creation: 15 May 2009
    Date of revision:
    Handle: RePEc:hub:wpecon:200909

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    Related research

    Keywords: Subexponential distributions; regular variation; O-regular variation; subordination;

    References

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    1. Ali, Mir M. & Mikhail, N. N. & Haq, M. Safiul, 1978. "A class of bivariate distributions including the bivariate logistic," Journal of Multivariate Analysis, Elsevier, vol. 8(3), pages 405-412, September.
    2. de Haan, L. & Omey, E. & Resnick, S., 1984. "Domains of attraction and regular variation in IRd," Journal of Multivariate Analysis, Elsevier, vol. 14(1), pages 17-33, February.
    3. Cline, Daren B. H. & Resnick, Sidney I., 1992. "Multivariate subexponential distributions," Stochastic Processes and their Applications, Elsevier, vol. 42(1), pages 49-72, August.
    4. Embrechts, Paul & Goldie, Charles M., 1982. "On convolution tails," Stochastic Processes and their Applications, Elsevier, vol. 13(3), pages 263-278, September.
    5. Omey, E. & Willekens, E., 1986. "Second order behaviour of the tail of a subordinated probability distribution," Stochastic Processes and their Applications, Elsevier, vol. 21(2), pages 339-353, February.
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