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Fixed Points of the Smoothing Transform: the Boundary Case


 
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1. Title Title of document Fixed Points of the Smoothing Transform: the Boundary Case
 
2. Creator Author's name, affiliation, country John D Biggins; University of Sheffield
 
2. Creator Author's name, affiliation, country Andreas E Kyprianou; Heriot-Watt University
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Smoothing transform; functional equation; branching random walk
 
3. Subject Subject classification 60J80 60G42
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
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7. Date (YYYY-MM-DD) 2005-06-13
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/255
 
10. Identifier Digital Object Identifier 10.1214/EJP.v10-255
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 10
 
12. Language English=en
 
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