Electronic Journal of Probability

Fixed Points of the Smoothing Transform: the Boundary Case

John Biggins and Andreas Kyprianou

Full-text: Open access

Abstract

Let $A=(A_1,A_2,A_3,\ldots)$ be a random sequence of non-negative numbers that are ultimately zero with $E[\sum A_i]=1$ and $E \left[\sum A_{i} \log A_i \right] \leq 0$. The uniqueness of the non-negative fixed points of the associated smoothing transform is considered. These fixed points are solutions to the functional equation $\Phi(\psi)= E \left[ \prod_{i} \Phi(\psi A_i) \right], $ where $\Phi$ is the Laplace transform of a non-negative random variable. The study complements, and extends, existing results on the case when $E\left[\sum A_{i} \log A_i \right]<0$. New results on the asymptotic behaviour of the solutions near zero in the boundary case, where $E\left[\sum A_{i} \log A_i \right]=0$, are obtained.

Article information

Source
Electron. J. Probab., Volume 10 (2005), paper no. 17, 609-631.

Dates
Accepted: 13 June 2005
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464816818

Digital Object Identifier
doi:10.1214/EJP.v10-255

Mathematical Reviews number (MathSciNet)
MR2147319

Zentralblatt MATH identifier
1110.60081

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G42: Martingales with discrete parameter

Keywords
Smoothing transform functional equation branching random walk

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Biggins, John; Kyprianou, Andreas. Fixed Points of the Smoothing Transform: the Boundary Case. Electron. J. Probab. 10 (2005), paper no. 17, 609--631. doi:10.1214/EJP.v10-255. https://projecteuclid.org/euclid.ejp/1464816818


Export citation