Implicit Renewal Theory and Power Tails on Trees

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 solutionsRtoand similar recursions, where (Q,N,C1,C2,…) is a nonnegative random vector withN∈ {0, 1, 2, 3,…} ∪ {∞}, andare independent and identically distributed copies ofR, independent of (Q,N,C1,C2,…); here ‘∨’ denotes the maximum operator.