Coupling on weighted branching trees

TL;DR

This paper develops bounds for the Kantorovich-Rubinstein distance between linear functions on weighted branching processes, with applications to random graph analysis and convergence of these processes.

Contribution

It introduces explicit bounds based on couplings for weighted branching processes and explores convergence conditions for linear functions, including cases with modified dependence structures.

Findings

Derived explicit bounds for Kantorovich-Rubinstein distance

Established convergence conditions for weighted branching processes

Analyzed fixed points of smoothing transformations

Abstract

This paper considers linear functions constructed on two different weighted branching processes and provides explicit bounds for their Kantorovich-Rubinstein distance in terms of couplings of their corresponding generic branching vectors. Motivated by applications to the analysis of random graphs, we also consider a variation of the weighted branching process where the generic branching vector has a different dependence structure from the usual one. By applying the bounds to sequences of weighted branching processes, we derive sufficient conditions for the convergence in the Kantorovich-Rubinstein distance of linear functions. We focus on the case where the limits are endogenous fixed points of suitable smoothing transformations.