Skip to main content

The Weighted Branching Process

  • Chapter
  • First Online:
Book cover Branching Processes and Their Applications

Part of the book series: Lecture Notes in Statistics ((LNSP,volume 219))

  • 1223 Accesses

Abstract

Branching processes have a lot of applications in the real world. Basically, one counts the number of offspring or individuals in a certain generation. But that is only part of the story. In reality the mother (parents) passes some value to the children and they to their children and so on. Examples of values are the DNA sequence, money, knowledge, influence and so on.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aldous, D., Bandyopadhyay, A.: A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15, 1047–1110 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alsmeyer, G.: The smoothing transform: a review of contraction results. In: Alsmeyer, G., Löwe, M. (eds.) Random Matrices and Iterated Random Functions. Springer Proceedings in Mathematics & Statistics, vol. 53, pp. 189–228. Springer, Berlin (2013)

    Chapter  Google Scholar 

  3. Alsmeyer, G., Kuhlbusch, D.: Double martingale structure and existence of f-moments for weighted branching processes. Preprint no. 08/05-S. Universität Münster, Germany (2006)

    Google Scholar 

  4. Alsmeyer, G., Meiners, M.: Fixed point of inhomogeneous smoothing transforms. arXiv: 1007.4509v2 (2011)

    Google Scholar 

  5. Alsmeyer, G., Meiners, M.: Fixed points of the smoothing transform: two-sided solutions. Probab. Theory Relat. Fields 155, 165–199 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alsmeyer, A., Roesler, U.: A stochastic fixed point equation related to weighted branching with deterministic weights. Electron J. Probab. 11, 27–56 (2005)

    Article  MathSciNet  Google Scholar 

  7. Alsmeyer, G., Roesler, U.: A stochastic fixed point equation for weighted minima and maxima. Ann. Henri. Poincaré 44, 89–103 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Alsmeyer, G., Biggins, J.D., Meiners, M.: The functional equation of the smoothing transform. Ann. Probab. 40, 2069–2105 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Athreya, K.B., Ney, P.E.: Branching Processes. Die Grundlehren der mathematischen Wissenschaften, vol. 196. Springer, Berlin (1972)

    Google Scholar 

  10. Attal, S.: Markov chains and dynamical systems: the open system point of view. arXiv:1010.2894 (2010)

    Google Scholar 

  11. Barnsley, M.F.: Fractals Everywhere, 2nd edn. AP Professional, New York (1993). ISBN-10: 0120790610, ISBN-13: 978-0120790616

    Google Scholar 

  12. Barnsley, M.F.: SuperFractals, 2nd edn. Cambridge University Press, Cambridge (2006). ISBN: 978-0521844932

    Book  MATH  Google Scholar 

  13. Barral, J., Mandelbrot, B.: Fractional multiplicative processes. Ann. Inst. Henri Poincaré Probab. Stat. 45, 1116–1129 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Biggins, J.D.: Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  15. Biggins, J.D., Kyprianou, A.E.: Measure change in multitype branching. Adv. Appl. Probab. 36, 544–481 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Burton, R., Roesler, U.: An L 2 convergence theorem for random affine mappings. J. App. Probab. 32, 183–192 (1995)

    Article  MathSciNet  Google Scholar 

  17. Durrett, R., Liggett, T.M.: Fixed points of the smoothing transform. Z. Wahrscheinlichkeitstheorie verw. Gebiete 64, 275–301 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  18. Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, New York (2003)

    Book  MATH  Google Scholar 

  19. Fill, J.A., Janson, S.: A characterization of the set of fixed points of the Quicksort transformation. Electron. Commun. Prob. 5, 77–84 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Grübel, R., Roesler, U. Asymptotic distribution theory for Hoare’s selection algorithm. Adv. Appl. Probab. 28, 252–269 (1996)

    Article  MATH  Google Scholar 

  21. Holley, R., Liggett, T.M.: Generalized potlatch and smoothing processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 55, 165–195 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  22. Iksanov, A., Meiners, M.: Fixed points of multivariate smoothing transforms with scalar weights. arXiv 1402.4147v1 (2014)

    Google Scholar 

  23. Kahane, J-P., Peyriére, J.: Sur Certaines Martingals de Benoît Mandelbrot. Adv. Math. 22, 131–145 (1976)

    Article  MATH  Google Scholar 

  24. Kellerer, H.G.: Ergodic behaviour of affine recursions I-III Preprints (1992)

    Google Scholar 

  25. Liu, Q.: Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Probab. 30, 85–112 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, Q.: On generalized multiplicative cascades. Stochastic Process. Appl. 86, 263–286 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lyons, R.: A simple path to Biggins’s martingale convergence for branching random walk. In: Athreya, K.B., Jagers, P. (eds.) Classical and Modern Branching Processes. IMA Volumes in Mathematics and its Applications, vol. 84, pp. 217–221. Springer, Berlin (1997)

    Chapter  Google Scholar 

  28. Neininger, R.: Probabilistic analysis of algorithms, stochastic fixed point equations and ideal metrics. Lecture Notes: Summer School on Probabilistic Methods in Combinatorics Graz, July 17–19 (2006)

    Google Scholar 

  29. Neininger, R., Rüschendorf, L.: General limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14, 378–418 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Neininger, R., Rüschendorf, L.: Analysis of algorithms by the contraction method: additive and max-recursive sequences. In: Interacting Stochastic Systems, pp. 435–450. Springer, Berlin (2005)

    Google Scholar 

  31. Neininger, R., Sulzbach, H.: On a functional contraction method. Ann. Probab. 43, 1777–1822 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pachter, L., Sturmfels, B.: Algebraic Statistics for Computational Biology. Cambridge University Press, Cambridge (2005). ISBN-13: 978-0521857000

    MathSciNet  MATH  Google Scholar 

  33. Rachev, S.T., Rüschendorf, L.: Probability metrics and recursive algorithms. Adv. Appl. Probab. 27, 770–799 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Régnier, M.: A limiting distribution for quicksort. RAIRO Theor. Inform. Appl. 23, 335–343 (1989)

    MathSciNet  MATH  Google Scholar 

  35. Roesler, U.: A limit theorem for “Quicksort”. RAIRO Theor. Inform. Appl. 25, 85–100 (1991)

    MathSciNet  Google Scholar 

  36. Roesler, U.: A fixed point theorem for distributions. Stoch. Process. Appl. 42, 195–214 (1992)

    Article  MathSciNet  Google Scholar 

  37. Roesler, U.: The weighted branching process. In: Dynamics of Complex and Irregular Systems (Bielefeld, 1991): Bielefeld Encounters in Mathematics and Physics VIII, pp. 154–165. World Scientific Publishing, River Edge, NJ (1993)

    Google Scholar 

  38. Roesler, U.: Almost sure convergence to the Quicksort process. Preprint (2015)

    Google Scholar 

  39. Roesler, U., Rüschendorf, L.: The contraction method for recursive algorithms. Algorithmica 29, 3–33 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rüschendorf, L.: On stochastic recursive equations of sum and max type. J. Appl. Probab. 43, 687–703 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  41. Waymire, E., Williams, S.: Markov cascades. In: Athreya, K.B., Jagers, P. (eds.) Classical and Modern Branching Processes, IMA Volumes in Mathematics and its Applications. vol. 84, pp. 305–321. Springer, Berlin (1997)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Uwe Roesler .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Roesler, U. (2016). The Weighted Branching Process. In: del Puerto, I., et al. Branching Processes and Their Applications. Lecture Notes in Statistics(), vol 219. Springer, Cham. https://doi.org/10.1007/978-3-319-31641-3_13

Download citation

Publish with us

Policies and ethics