Abstract
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 =D ∑i=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}i∈N} are independent and identically distributed copies of R, independent of (Q, N, C1, C2,...); here '∨' denotes the maximum operator.
Citation
Predrag R. Jelenković. Mariana Olvera-Cravioto. "Implicit renewal theory and power tails on trees." Adv. in Appl. Probab. 44 (2) 528 - 561, June 2012. https://doi.org/10.1239/aap/1339878723
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