Advances in Applied Probability

Implicit renewal theory and power tails on trees

Predrag R. Jelenković and Mariana Olvera-Cravioto

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We extend Goldie's (1991) implicit renewal theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power-tail asymptotics of the distributions of the solutions R to R =Di=1N Ci Ri + Q, R =D (∨i=1N Ci Ri) ∨Q, and similar recursions, where (Q, N, C1, C2,...) is a nonnegative random vector with N ∈ {0, 1, 2, 3,...} ∪ {∞}, and {Ri}iN} are independent and identically distributed copies of R, independent of (Q, N, C1, C2,...); here '∨' denotes the maximum operator.

Article information

Adv. in Appl. Probab., Volume 44, Number 2 (2012), 528-561.

First available in Project Euclid: 16 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H25: Random operators and equations [See also 47B80]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F10: Large deviations 60K05: Renewal theory

Implicit renewal theory weighted branching process multiplicative cascade stochastic recursion power law large deviations stochastic fixed-point equation


Jelenković, Predrag R.; Olvera-Cravioto, Mariana. Implicit renewal theory and power tails on trees. Adv. in Appl. Probab. 44 (2012), no. 2, 528--561. doi:10.1239/aap/1339878723.

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