Implicit Renewal Theory and Power Tails on Trees

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, C_1,..., C_N) is a nonnegative random vector with N in {0, 1, 2, 3, ..., infinity}, and {R_i}_{i >= 1} are iid copies of R, independent of (Q, N, C_1,..., C_N); =_D denotes the equality in distribution.