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Original Articles

Fixed points of inhomogeneous smoothing transforms

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Pages 1287-1304 | Received 10 Aug 2010, Accepted 12 May 2011, Published online: 06 Sep 2011

References

  • G. Alsmeyer, J. Biggins, and M. Meiners, The functional equation of the smoothing transform (2010), to appear in Ann. Probab. Preprint available at www.arXiv.org: 0906.3133v2 [math.PR]
  • Alsmeyer , G. and Meiners , M. 2009 . A min-type stochastic fixed-point equation related to the smoothing transformation . Theory Stoch. Proc. , 15 ( 31 ) : 19 – 41 .
  • Alsmeyer , G. and Rösler , U. 2005 . A stochastic fixed point equation related to weighted branching with deterministic weights . Electron. J. Probab. , 11 ( 2 ) : 27 – 56 . (electronic)
  • Athreya , K.B. and Ney , P.E. 1972 . Branching Processes, Die Grundlehren der mathematischen Wissenschaften, Band 196 , New York : Springer-Verlag .
  • Biggins , J.D. 1998 . Lindley-type equations in the branching random walk . Stoch. Proc. Appl. , 75 : 105 – 133 .
  • Biggins , J.D. 1977 . Martingale convergence in the branching random walk . J. Appl. Probab. , 14 : 25 – 37 .
  • Biggins , J.D. and Kyprianou , A.E. 1997 . Seneta-heyde norming in the branching random walk . Ann. Probab. , 25 : 337 – 360 .
  • Durrett , R. and Liggett , T.M. 1983 . Fixed points of the smoothing transformation . Z. Wahrsch. Verw. Gebiete , 64 : 275 – 301 .
  • Feller , W. 1971 . An Introduction to Probability Theory and Its Applications, Vol. II , Second edition , New York : John Wiley & Sons, Inc. .
  • Goldie , C.M. and Maller , R.A. 2000 . Stability of perpetuities . Ann. Probab. , 28 : 1195 – 1218 .
  • Hu , Y. and Shi , Z. 2009 . Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees . Ann. Probab. , 37 : 742 – 789 .
  • Jelenković , P. and Olvera-Cravioto , M. 2010 . Implicit renewal theory and power tails on trees . Available at www.arxiv.org: 1006.3295v2
  • Jelenković , P. and Olvera-Cravioto , M. 2010 . Information ranking and power laws on trees . Adv. Appl. Probab. , 42 : 1057 – 1093 .
  • Liu , Q. 1998 . Fixed points of a generalized smoothing transformation and applications to the branching random walk . Adv. Appl. Probab. , 30 : 85 – 112 .
  • Lyons , R. 1997 . “ A simple path to Biggins' martingale convergence for branching random walk ” . In Classical and Modern Branching Processes , New York : Springer . (Minneapolis, MN, 1994), Volume 84 of IMA Vol. Math. Appl. pp. 217–221
  • Nerman , O. 1981 . On the convergence of supercritical general (C-M-J) branching processes . Z. Wahrsch. Verw. Gebiete , 57 : 365 – 395 .
  • L. Page and S. Brin, The anatomy of a large-scale hypertextual web search engine. Proceedings of the Seventh International Web Conference (WWW 98), Brisbane (1998)
  • J. Spitzmann, Lösungen inhomogener stochastischer Fixpunktgleichungen (Solutions of inhomogeneous stochastic fixed-point equations), Ph.D. thesis, Christian-Albrechts-Universität zu Kiel, Mathematisches Seminar (2010)
  • Takács , L. 1962 . “ Introduction to the theory of queues ” . In University Texts in the Mathematical Sciences , New York : Oxford University Press .
  • Volkovich , Y. and Litvak , N. 2010 . Asymptotic analysis for personalized web search . Adv. Appl. Probab. , 42 : 577 – 604 .

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