Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

On a Stochastic Difference Equation and a Representation of Non-Negative Infinitely Divisible Random Variables

Wim Vervaat
Advances in Applied Probability
Vol. 11, No. 4 (Dec., 1979), pp. 750-783
DOI: 10.2307/1426858
Stable URL: http://www.jstor.org/stable/1426858
Page Count: 34
  • Read Online (Free)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On a Stochastic Difference Equation and a Representation of Non-Negative Infinitely Divisible Random Variables
Preview not available

Abstract

The present paper considers the stochastic difference equation Yn=AnYn-1+Bn with i.i.d. random pairs (An,Bn) and obtains conditions under which Yn converges in distribution. This convergence is related to the existence of solutions of $Y\overset \text{d}\to{=}AY+B,\ Y$ and (A,B) independent, and the convergence w.p. 1 of ∑ A1A2⋯ An-1Bn. A second subject is the series ∑ Cnf(Tn) with (Cn) a sequence of i.i.d. random variables, (Tn) the sequence of points of a Poisson process and f a Borel function on (0,∞). The resulting random variable turns out to be infinitely divisible, and its Lévy-Hinčin representation is obtained. The two subjects coincide in case An and Cn are independent, Bn=AnCn, An=Un 1/α with Un a uniform random variable, f(x)=e-x/α.

Page Thumbnails

  • Thumbnail: Page 
750
    750
  • Thumbnail: Page 
751
    751
  • Thumbnail: Page 
752
    752
  • Thumbnail: Page 
753
    753
  • Thumbnail: Page 
754
    754
  • Thumbnail: Page 
755
    755
  • Thumbnail: Page 
756
    756
  • Thumbnail: Page 
757
    757
  • Thumbnail: Page 
758
    758
  • Thumbnail: Page 
759
    759
  • Thumbnail: Page 
760
    760
  • Thumbnail: Page 
761
    761
  • Thumbnail: Page 
762
    762
  • Thumbnail: Page 
763
    763
  • Thumbnail: Page 
764
    764
  • Thumbnail: Page 
765
    765
  • Thumbnail: Page 
766
    766
  • Thumbnail: Page 
767
    767
  • Thumbnail: Page 
768
    768
  • Thumbnail: Page 
769
    769
  • Thumbnail: Page 
770
    770
  • Thumbnail: Page 
771
    771
  • Thumbnail: Page 
772
    772
  • Thumbnail: Page 
773
    773
  • Thumbnail: Page 
774
    774
  • Thumbnail: Page 
775
    775
  • Thumbnail: Page 
776
    776
  • Thumbnail: Page 
777
    777
  • Thumbnail: Page 
778
    778
  • Thumbnail: Page 
779
    779
  • Thumbnail: Page 
780
    780
  • Thumbnail: Page 
781
    781
  • Thumbnail: Page 
782
    782
  • Thumbnail: Page 
783
    783