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Limiting distributions of two random sequences

Author

Listed:
  • Chen, Robert
  • Goodman, Richard
  • Zame, Alan

Abstract

For fixed p (0 = 2, with probability p let {Ln, Rn} = {Ln - 1, Xn} or = {Xn, Rn - 1} according as , with probability 1 - p let {Ln, Rn} = {Xn, Rn - 1} or = {Ln - 1, Xn} according as , and let Xn + 1 be a uniform random variable over {Ln, Rn}. By this iterated procedure, a random sequence {Xn}n >= 1 is constructed, and it is easy to see that Xn converges to a random variable Yp (say) almost surely as n --> [infinity]. Then what is the distribution of Yp? It is shown that the Beta, (2, 2) distribution is the distribution of Y1; that is, the probability density function of Y1 is g(y) = 6y(1 - y) I0,1(y). It is also shown that the distribution of Y0 is not a known distribution but has some interesting properties (convexity and differentiability).

Suggested Citation

  • Chen, Robert & Goodman, Richard & Zame, Alan, 1984. "Limiting distributions of two random sequences," Journal of Multivariate Analysis, Elsevier, vol. 14(2), pages 221-230, April.
    • Handle: RePEc:eee:jmvana:v:14:y:1984:i:2:p:221-230
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    Citations

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    Cited by:

    1. McKinlay, Shaun, 2017. "On beta distributed limits of iterated linear random functions," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 33-41.
    2. Johnson, Norman L. & Kotz, Samuel, 1995. "Use of moments in studies of limit distributions arising from iterated random subdivisions of an interval," Statistics & Probability Letters, Elsevier, vol. 24(2), pages 111-119, August.
    3. Margarete Knape & Ralph Neininger, 2008. "Approximating Perpetuities," Methodology and Computing in Applied Probability, Springer, vol. 10(4), pages 507-529, December.

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