Convergence Rates in the Implicit Renewal Theorem on Trees

R D = f(Ci, Ri, 1 ≤ i ≤ N), where f(·) is a possibly random real valued function, N ∈ {0, 1, 2, 3, . . . } ∪ {∞}, {Ci}i∈N are real valued random weights and {Ri}i∈N are iid copies of R, independent of (N,C1, C2, . . . ); D = represents equality in distribution. In the recent paper [10], an Implicit Renewal Theorem was developed that enables the characterization of the power tail asymptotics of the solutions R to many equations that fall into this category. In this paper we complement the analysis in [10] to provide the corresponding rate of convergence.