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2005 Fixed Points of the Smoothing Transform: the Boundary Case
John Biggins, Andreas Kyprianou
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Electron. J. Probab. 10: 609-631 (2005). DOI: 10.1214/EJP.v10-255

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.

Citation

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John Biggins. Andreas Kyprianou. "Fixed Points of the Smoothing Transform: the Boundary Case." Electron. J. Probab. 10 609 - 631, 2005. https://doi.org/10.1214/EJP.v10-255

Information

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

zbMATH: 1110.60081
MathSciNet: MR2147319
Digital Object Identifier: 10.1214/EJP.v10-255

Subjects:
Primary: 60G42, 60J80

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