Advances in Applied Probability

Coupling on weighted branching trees

Ningyuan Chen and Mariana Olvera-Cravioto

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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.

Article information

Source
Adv. in Appl. Probab., Volume 48, Number 2 (2016), 499-524.

Dates
First available in Project Euclid: 9 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.aap/1465490760

Mathematical Reviews number (MathSciNet)
MR3511773

Zentralblatt MATH identifier
1343.60127

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60B10: Convergence of probability measures 60H25: Random operators and equations [See also 47B80]

Keywords
Weighted branching processes smoothing transform coupling Kantorovich–Rubinstein distance Wasserstein distance weak convergence

Citation

Chen, Ningyuan; Olvera-Cravioto, Mariana. Coupling on weighted branching trees. Adv. in Appl. Probab. 48 (2016), no. 2, 499--524. https://projecteuclid.org/euclid.aap/1465490760


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References

  • Alsmeyer, G. and Iksanov, A. (2009). A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks. Electron. J. Prob. 14, 289–313.
  • Alsmeyer, G. and Meiners, M. (2012). Fixed points of inhomogeneous smoothing transforms. J. Differ. Equ. Appl. 18, 1287–1304.
  • Alsmeyer, G. and Meiners, M. (2013). Fixed points of the smoothing transform: two-sided solutions. Prob. Theory Relat. Fields 155, 165–199.
  • Alsmeyer, G., Biggins, J. D. and Meiners, M. (2012). The functional equation of the smoothing transform. Ann. Prob. 40, 2069–2105.
  • Bickel, P. J. and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9, 1196–1217.
  • Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37.
  • Bollobás, B. (2001). Random Graphs, 2nd edn. Cambridge University Press.
  • Chen, N. and Olvera-Cravioto, M. (2013). Directed random graphs with given degree distributions. Stoch. Systems 3, 147–186.
  • Chen, N., Litvak, N. and Olvera-Cravioto, M. (2014). Ranking algorithms on directed configuration networks. Preprint. Available at http://arxiv.org/abs/1409.7443.
  • Durrett, R. (2010). Probability: Theory and Examples, 4th edn. Cambridge University Press.
  • Fill, J. A. and Janson, S. (2001). Approximating the limiting Quicksort distribution. Random Structures Algorithms 19, 376–406.
  • Iksanov, A. M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Process. Appl. 114, 27–50.
  • Iksanov, A. and Meiners, M. (2015). Fixed points of multivariate smoothing transforms with scalar weights. ALEA Latin Amer. J. Prob. Math. Statist. 12, 69–114.
  • Jelenković, P. R. and Olvera-Cravioto, M. (2010). Information ranking and power laws on trees. Adv. Appl. Prob. 42, 1057–1093.
  • Jelenković, P. R. and Olvera-Cravioto, M. (2012). Implicit renewal theorem for trees with general weights. Stoch. Process. Appl. 122, 3209–3238.
  • Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Prob. 30, 85–112.
  • Liu, Q. (2000). On generalized multiplicative cascades. Stoch. Process. Appl. 86, 263–286.
  • Lyons, R. (1997). A simple path to Biggins' martingale convergence for branching random walk. In Classical and Modern Branching Processes, Springer, New York, pp. 217–221.
  • Neininger, R. (2001). On a multivariate contraction method for random recursive structures with applications to Quicksort. Random Structures Algorithms 19, 498–524.
  • Olvera-Cravioto, M. (2012). Tail behavior of solutions of linear recursions on trees. Stoch. Process. Appl. 122, 1777–1807.
  • Rösler, U. (1991). A limit theorem for “Quicksort”. RAIRO Inf. Théor. Appl. 25, 85–100.
  • Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica 29, 3–33.
  • van der Hofstad, R. (2014). Random graphs and complex networks. Preprint. Available at http://www.win. tue.nl/$\sim$rhofstad/NotesRGCN.html.
  • van der Hofstad, R., Hooghiemstra, G. and Van Mieghem, P. (2005). Distances in random graphs with finite variance degrees. Random Structures Algorithms 27, 76–123.
  • Villani, C. (2009). Optimal Transport: Old and New. Springer, Berlin.
  • Volkovich, Y. and Litvak, N. (2010). Asymptotic analysis for personalized web search. Adv. Appl. Prob. 42, 577–604.
  • Williams, D. (1991). Probability with Martingales. Cambridge University Press.