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Discounted branching random walks

Published online by Cambridge University Press:  01 July 2016

K. B. Athreya
K. B. Athreya*
Affiliation:
Iowa State University
*
∗Postal address: Departments of Mathematics and Statistics, Iowa State University, Ames, IA 50011, USA.
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Abstract

Let F(·) be a c.d.f. on [0,∞), f(s) = ∑∞0pjsi a p.g.f. with p0 = 0, < 1 < m = Σjpj < ∞ and 1 < ρ <∞. For the functional equation for a c.d.f. H(·) on [0,∞] we establish that if 1 – F(x) = O(x–θ) for some θ > α =(log m)/(log p) there exists a unique solution H(·) to (∗) in the class C of c.d.f.’s satisfying 1 – H(x) = o(x–α).

We give a probabilistic construction of this solution via branching random walks with discounting. We also show non-uniqueness if the condition 1 – H(x) = o(x–α) is relaxed.

Keywords

FUNCTIONAL EQUATIONDISCOUNTING
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 1 , March 1985 , pp. 53 - 66
Copyright
Copyright © Applied Probability Trust 1985 

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Footnotes

Research supported in part by NSF grant MCS-8201456 and by SHRI of ISU.

References

1. Asmussen, S. and Kaplan, N. (1976) Branching random walks I & II. Stoch. Proc. Appl. 4, 1–31.Google Scholar
2. Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
3. Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
4. Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, New York.Google Scholar