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Coupling on weighted branching trees

Part of: Stochastic analysis Markov processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  10 June 2016

Ningyuan Chen  and
Mariana Olvera-Cravioto
Ningyuan Chen*
Affiliation:
Columbia University
Mariana Olvera-Cravioto*
Affiliation:
Columbia University
*
* Postal address: Industrial Engineering and Operations Research Department, Columbia University, 321 S. W. Mudd Building, 500 W. 120th Street, New York, NY 10027, USA.
* Postal address: Industrial Engineering and Operations Research Department, Columbia University, 321 S. W. Mudd Building, 500 W. 120th Street, New York, NY 10027, USA.
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Abstract

In this paper we consider linear functions constructed on two different weighted branching processes and provide 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.

Keywords

Weighted branching processessmoothing transformcouplingKantorovich–Rubinstein distanceWasserstein distanceweak convergence

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60B10: Convergence of probability measures 60H25: Random operators and equations
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 2 , June 2016 , pp. 499 - 524
Copyright
Copyright © Applied Probability Trust 2016 

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