Coupling on weighted branching trees

Ningyuan Chen, Mariana Olvera-Cravioto

Published 2014-10-04Version 1

This paper considers linear functions constructed on two different weighted branching trees and provides explicit bounds for their Kantorovich-Rubinstein distance in terms of couplings of their corresponding generic branching vectors. By applying these bounds to a sequence of weighted branching trees, we derive the weak convergence of the corresponding linear processes. In the special case where sequence of trees converges to a weighted branching process, the limits can be represented as the endogenous solution to a stochastic fixed-point equation of the form $$R \stackrel{D}{=} \sum_{i=1}^N C_i R_i + Q,$$ where $(Q, N, \{C_i\})$ is a real-valued random vector with $N \in \mathbb{N} = \{0, 1, 2, \dots\}$ and $\{R_i\}_{i \in \mathbb{N}}$ is a sequence of i.i.d. copies of $R$, independent of $(Q, N, \{ C_i \})$. The type of assumptions imposed on the generic branching vectors are suitable for applications in the analysis of a variety of random graph models.