Implicit Renewal Theory and Power Tails on Trees

TL;DR

This paper extends Goldie's Implicit Renewal Theorem to analyze power tail behaviors of solutions to recursive distributional equations on weighted branching trees, providing a new method for deriving tail asymptotics.

Contribution

It develops a generalized renewal theorem for recursive equations on trees, enabling the analysis of power tail asymptotics for complex recursions involving sums and maxima.

Findings

Derived power tail asymptotics for solutions to recursive equations on trees.

Extended the Implicit Renewal Theorem to weighted branching structures.

Provided a framework for analyzing tail behaviors in recursive stochastic models.

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, 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.