Implicit Renewal Theory and Power Tails on Trees

2010/06/16 by Predrag R. Jelenković, Jelenković, Predrag R., Mariana Olvera‐Cravioto +1 · 1 citation
Mathematics · Physics and Astronomy · #60F10 #60H25 (Primary) 60J80 #60K05 (Secondary) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Performance (cs.PF) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1006.3295

openalex publication_date 2010/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 sum_i=1^N C_i R_i + Q, R =_D max(max_i=1^N C_i R_i, Q), and similar recursions, where (Q, N, C₁,..., C_N) is a nonnegative random vector with N in 0, 1, 2, 3, ..., infinity, and R_{ii >= 1} are iid copies of R, independent of (Q, N, C₁,..., C_N); =_D denotes the equality in distribution.

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