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Implicit Renewal Theory and Power Tails on Trees

  • Predrag R. Jelenković (a1) and Mariana Olvera-Cravioto (a1)

Abstract

We extend Goldie's (1991) implicit renewal theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power-tail asymptotics of the distributions of the solutions R to and similar recursions, where (Q, N, C 1, C 2,…) is a nonnegative random vector with N ∈ {0, 1, 2, 3,…} ∪ {∞}, and are independent and identically distributed copies of R, independent of (Q, N, C 1, C 2,…); here ‘∨’ denotes the maximum operator.

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Copyright

Corresponding author

Postal address: Department of Electrical Engineering, Columbia University, New York, NY 10027, USA.
∗∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA. Email address: molvera@ieor.columbia.edu

Footnotes

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Supported by the NSF, grant no. CMMI-1131053.

Supported by the NSF, grant no. CMMI-1131053.

Footnotes

References

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[1] Aldous, D. J. and Bandyopadhyay, A. (2005). A survey of max-type recursive distributional equation. Ann. Appl. Prob. 15, 10471110.
[2] Alsmeyer, G. and Kuhlbusch, D. (2010). Double martingale structure and existence of ϕ-moments for weighted branching processes. Münster J. Math. 3, 163212.
[3] Alsmeyer, G. and Meiners, M. (2012). Fixed points of inhomogeneous smoothing transforms. To appear in J. Diff. Equat. Appl.
[4] Alsmeyer, G. and Rösler, U. (2005). A stochastic fixed point equation related to weighted branching with deterministic weights. Electron. J. Prob. 11, 2756.
[5] Alsmeyer, G. and Rösler, U. (2008). A stochastic fixed point equation for weighted minima and maxima. Ann. Inst. H. Poincaré Prob. Statist. 44, 89103.
[6] Alsmeyer, G., Biggins, J. D. and Meiners, M. (2012). The functional equation of the smoothing transform. To appear in Ann. Prob.
[7] Athreya, K. B., McDonald, D. and Ney, P. (1978). Limit theorems for semi-Markov processes and renewal theory for Markov chains. Ann. Prob. 6, 788797.
[8] Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 2537.
[9] Biggins, J. D. and Kyprianou, A. E. (1997). Seneta–Heyde norming in the branching random walk. Ann. Prob. 25, 337360.
[10] Billingsley, P. (1995). Probability and Measure, 3rd edn. Wiley-Interscience, New York.
[11] Brandt, A. (1986). The stochastic equation Y n+1 = A n Y n + B n with stationary coefficients. Adv. Appl. Prob. 18, 211220.
[12] Durret, R. and Liggett, T. M. (1983). Fixed points of the smoothing transformation. Z. Wahrscheinlichkeitsth. 64, 275301.
[13] Fill, J. A. and Janson, S. (2001). Approximating the limiting Quicksort distribution. Random Structures Algorithms 19, 376406.
[14] Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126166.
[15] Grincevičius, A. K. (1975). One limit distribution for a random walk on the line. Lithuanian Math. J. 15, 580589.
[16] Holley, R. and Liggett, T. M. (1981). Generalized potlatch and smoothing processes. Z. Wahrscheinlichkeitsth. 55, 165195.
[17] Iksanov, A. M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Process. Appl. 114, 2750.
[18] Jagers, P. and Rösler, U. (2004). Stochastic fixed points for the maximum. In Mathematics and Computer Science III, eds Drmota, M. et al., Birkhäuser, Basel, pp. 325338.
[19] Jelenković, P. R. and Olvera-Cravioto, M. (2010). Information ranking and power laws on trees. Adv. Appl. Prob. 42, 10571093.
[20] Kahane, J.-P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, 131145.
[21] Kesten, H. (1973). Random difference equations and renewal theory for product of random matrices. Acta Math. 131, 207248.
[22] Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Prob. 30, 85112.
[23] Liu, Q. (2000). On generalized multiplicative cascades. Stoch. Process. Appl. 86, 263286.
[24] Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Prob. 14, 378418.
[25] Rösler, U. (1993). The weighted branching process. In Dynamics of Complex and Irregular Systems (Bielefeld, 1991; Bielefeld Encount. Math. Phys. VIII), World Scientific, River Edge, NJ, pp. 154165.
[26] Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica 29, 333.
[27] Volkovich, Y. and Litvak, N. (2010). Asymptotic analysis for personalized web search. Adv. Appl. Prob. 42, 577604.
[28] Waymire, E. C. and Williams, S. C. (1995). Multiplicative cascades: dimension spectra and dependence. J. Fourier Anal. Appl. 1995, 589609.

Keywords

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Implicit Renewal Theory and Power Tails on Trees

  • Predrag R. Jelenković (a1) and Mariana Olvera-Cravioto (a1)

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