Implicit renewal theorem for trees with general weights

Stochastic Processes and their Applications

Volumn 122, Issue 9, 2012, Pages 3209-3238

  Jelenković, Predrag R ^a   Olvera Cravioto, Mariana ^a  

^a Department of Earth and Environmental Sciences   (United States)

Author keywords

Implicit renewal theory; Large deviations; Multiplicative cascades; Power laws; Smoothing transforms; Stochastic fixed point equations; Stochastic recursions; Weighted branching processes

Indexed keywords

BRANCHING PROCESS; FIXED POINT EQUATION; LARGE DEVIATIONS; MULTIPLICATIVE CASCADES; POWER-LAW; RECURSIONS; RENEWAL THEORY;

COMPUTER SIMULATION; STATISTICS;

STOCHASTIC SYSTEMS;

EID: 84862667136     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2012.05.004     Document Type: Article

References (27)

1. D.J. Aldous, and A. Bandyopadhyay A survey of max-type recursive distributional equation Ann. Appl. Probab. 15 2 2005 1047 1110
2. G. Alsmeyer, J.D. Biggins, M. Meiners, The functional equation of the smoothing transform, Ann. Probab. arXiv:0906.3133, 2012 (in press).
3. G. Alsmeyer, and D. Kuhlbusch Double martingale structure and existence of φ-moments for weighted branching processes Münster J. Math. 1 2008
4. G. Alsmeyer, M. Meiners, Fixed points of the smoothing transform: two-sided solutions, Probab. Theory Relat. Fields, arXiv:1009.2412, 2012 (in press).
5. G. Alsmeyer, and U. Rösler A stochastic fixed point equation related to weighted branching with deterministic weights Electron. J. Probab. 11 2005 27 56 (Pubitemid 43157372)
6. J.D. Biggins Martingale convergence in the branching random walk J. Appl. Probab. 14 1 1977 25 37
7. J.D. Biggins, and A.E. Kyprianou Seneta-Heyde norming in the branching random walk Ann. Probab. 25 1 1997 337 360
8. Y.S. Chow, and H. Teicher Probability Theory 1988 Springer-Verlag New York
9. R. Durret, and T. Liggett Fixed points of the smoothing transformation Z. Wahrscheinlichkeitstheor. Verwandte Geb. 64 1983 275 301
10. J.A. Fill, and S. Janson Approximating the limiting Quicksort distribution Random Structures Algorithms 19 3-4 2001 376 406
11. C.M. Goldie Implicit renewal theory and tails of solutions of random equations Ann. Appl. Probab. 1 1 1991 126 166
12. A.K. Grincevičius One limit distribution for a random walk on the line Lith. Math. J. 15 1975 580 589
13. R. Holley, and T. Liggett Generalized potlatch and smoothing processes Z. Wahrscheinlichkeitstheor. Verwandte Geb. 55 1981 165 195
14. A.M. Iksanov Elementary fixed points of the BRW smoothing transforms with infinite number of summands Stochastic Process. Appl. 114 2004 27 50 (Pubitemid 39305452)
15. P.R. Jelenković, and M. Olvera-Cravioto Information ranking and power laws on trees Adv. in Appl. Probab. 42 4 2010 1057 1093
16. P.R. Jelenković, and M. Olvera-Cravioto Implicit renewal theory and power tails on trees Adv. Appl. Probab. 44 2 2012 528 561
17. J.P. Kahane, and J. Peyrière Sur certaines martingales de Benoit Mandelbrot Adv. Math. 22 1976 131 145
18. H. Kesten Random difference equations and renewal theory for products of random matrices Acta Math. 131 1 1973 207 248
19. Q. Liu Fixed points of a generalized smoothing transformation and applications to the branching random walk Adv. in Appl. Probab. 30 1998 85 112 (Pubitemid 128555210)
20. Q. Liu On generalized multiplicative cascades Stochastic Process. Appl. 86 2000 263 286
21. P. Negadailov, Limit theorems for random recurrences and renewal-type processes, Ph.D. Thesis, 2010. Available at: http://igitur-archive.library.uu. nl/dissertations/.
22. R. Neininger, and L. Rüschendorf A general limit theorem for recursive algorithms and combinatorial structures Ann. Appl. Probab. 14 1 2004 378 418 (Pubitemid 41449261)
23. U. Rösler The weighted branching process Dynamics of Complex and Irregular Systems Bielefeld, 1991 Bielefeld Encounters in Mathematics and Physics vol. VIII 1993 World Science Publishing River Edge, NJ 154 165
24. U. Rösler, and L. Rüschendorf The contraction method for recursive algorithms Algorithmica 29 1-2 2001 3 33 (Pubitemid 33711214)
25. M.S. Sgibnev The matrix analogue of the Blackwell renewal theorem on the real line Sb. Math. 197 3 2006 369 386 (Pubitemid 44218478)
26. Y. Volkovich, and N. Litvak Asymptotic analysis for personalized web search Adv. in Appl. Probab. 42 2 2010 577 604
27. B. Zwart Tail asymptotics for the busy period in the GI/G/1 queue Math. Oper. Res. 26 3 2001 485 493