Implicit Renewal Theory and Power Tails on Trees

This is the authors' abstract. We don't add key points for this paper.

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