Implicit renewal theory and power tails on trees
Predrag R. Jelenković and Mariana Olvera-Cravioto
Source: Adv. in Appl. Probab. Volume 44, Number 2
(2012), 528-561.
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 R =D ∑i=1N Ci Ri + Q, R =D (∨i=1N Ci Ri) ∨Q, and similar recursions, where (Q, N, C1, C2,...) is a nonnegative random vector with N ∈ {0, 1, 2, 3,...} ∪ {∞}, and {Ri}i∈N} are independent and identically distributed copies of R, independent of (Q, N, C1, C2,...); here '∨' denotes the maximum operator.
First Page:
Show
Hide
Keywords: Implicit renewal theory; weighted branching process; multiplicative cascade; stochastic recursion; power law; large deviations; stochastic fixed-point equation
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aap/1339878723
Digital Object Identifier: doi:10.1239/aap/1339878723
Zentralblatt MATH identifier: 06055133
Mathematical Reviews number (MathSciNet): MR2977407
References
Aldous, D. J. and Bandyopadhyay, A. (2005). A survey of max-type recursive distributional equation. Ann. Appl. Prob. 15, 1047–1110.
Mathematical Reviews (MathSciNet): MR2134098
Zentralblatt MATH: 1105.60012
Digital Object Identifier: doi:10.1214/105051605000000142
Project Euclid: euclid.aoap/1115137969
Alsmeyer, G. and Kuhlbusch, D. (2010). Double martingale structure and existence of $\phi$-moments for weighted branching processes. Münster J. Math. 3, 163–212.
Mathematical Reviews (MathSciNet): MR2775360
Alsmeyer, G. and Meiners, M. (2012). Fixed points of inhomogeneous smoothing transforms. To appear in J. Diff. Equat. Appl.
Alsmeyer, G. and Rösler, U. (2005). A stochastic fixed point equation related to weighted branching with deterministic weights. Electron. J. Prob. 11, 27–56.
Mathematical Reviews (MathSciNet): MR2199054
Zentralblatt MATH: 1110.60080
Digital Object Identifier: doi:10.1214/EJP.v11-296
Alsmeyer, G. and Rösler, U. (2008). A stochastic fixed point equation for weighted minima and maxima. Ann. Inst. H. Poincaré Prob. Statist. 44, 89–103.
Mathematical Reviews (MathSciNet): MR2451572
Zentralblatt MATH: 1176.60006
Digital Object Identifier: doi:10.1214/07-AIHP104
Project Euclid: euclid.aihp/1203969869
Alsmeyer, G., Biggins, J. D. and Meiners, M. (2012). The functional equation of the smoothing transform. To appear in Ann. Prob.
Athreya, K. B., McDonald, D. and Ney, P. (1978). Limit theorems for semi-Markov processes and renewal theory for Markov chains. Ann. Prob. 6, 788–797.
Mathematical Reviews (MathSciNet): MR503952
Zentralblatt MATH: 0397.60052
Digital Object Identifier: doi:10.1214/aop/1176995429
Project Euclid: euclid.aop/1176995429
Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37.
Mathematical Reviews (MathSciNet): MR433619
Zentralblatt MATH: 0356.60053
Digital Object Identifier: doi:10.2307/3213258
Biggins, J. D. and Kyprianou, A. E. (1997). Seneta–Heyde norming in the branching random walk. Ann. Prob. 25, 337–360.
Mathematical Reviews (MathSciNet): MR1428512
Zentralblatt MATH: 0873.60062
Digital Object Identifier: doi:10.1214/aop/1024404291
Project Euclid: euclid.aop/1024404291
Billingsley, P. (1995). Probability and Measure, 3rd edn. Wiley-Interscience, New York.
Mathematical Reviews (MathSciNet): MR1324786
Brandt, A. (1986). The stochastic equation $Y_{n+1} = A_n Y_n + B_n$ with stationary coefficients. Adv. Appl. Prob. 18, 211–220.
Mathematical Reviews (MathSciNet): MR827336
Zentralblatt MATH: 0588.60056
Digital Object Identifier: doi:10.2307/1427243
Durret, R. and Liggett, T. M. (1983). Fixed points of the smoothing transformation. Z. Wahrscheinlichkeitsth. 64, 275–301.
Mathematical Reviews (MathSciNet): MR716487
Fill, J. A. and Janson, S. (2001). Approximating the limiting Quicksort distribution. Random Structures Algorithms 19, 376–406.
Mathematical Reviews (MathSciNet): MR1871560
Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.
Mathematical Reviews (MathSciNet): MR1097468
Zentralblatt MATH: 0724.60076
Digital Object Identifier: doi:10.1214/aoap/1177005985
Project Euclid: euclid.aoap/1177005985
Grincevičius, A. K. (1975). One limit distribution for a random walk on the line. Lithuanian Math. J. 15, 580–589.
Holley, R. and Liggett, T. M. (1981). Generalized potlatch and smoothing processes. Z. Wahrscheinlichkeitsth. 55, 165–195.
Mathematical Reviews (MathSciNet): MR608015
Iksanov, A. M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Process. Appl. 114, 27–50.
Mathematical Reviews (MathSciNet): MR2094146
Zentralblatt MATH: 1081.60017
Digital Object Identifier: doi:10.1016/j.spa.2004.06.002
Jagers, P. and Rösler, U. (2004). Stochastic fixed points for the maximum. In Mathematics and Computer Science III, eds M. Drmota et al., Birkhäuser, Basel, pp. 325–338.
Mathematical Reviews (MathSciNet): MR2090523
Digital Object Identifier: doi:10.1007/978-3-0348-7915-6_34
Jelenković, P. R. and Olvera-Cravioto, M. (2010). Information ranking and power laws on trees. Adv. Appl. Prob. 42, 1057–1093.
Mathematical Reviews (MathSciNet): MR2796677
Zentralblatt MATH: 1211.60026
Digital Object Identifier: doi:10.1239/aap/1293113151
Project Euclid: euclid.aap/1293113151
Kahane, J.-P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, 131–145.
Mathematical Reviews (MathSciNet): MR431355
Zentralblatt MATH: 0349.60051
Digital Object Identifier: doi:10.1016/0001-8708(76)90151-1
Kesten, H. (1973). Random difference equations and renewal theory for product of random matrices. Acta Math. 131, 207–248.
Mathematical Reviews (MathSciNet): MR440724
Zentralblatt MATH: 0291.60029
Digital Object Identifier: doi:10.1007/BF02392040
Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Prob. 30, 85–112.
Mathematical Reviews (MathSciNet): MR1618888
Zentralblatt MATH: 0909.60075
Digital Object Identifier: doi:10.1239/aap/1035227993
Project Euclid: euclid.aap/1035227993
Liu, Q. (2000). On generalized multiplicative cascades. Stoch. Process. Appl. 86, 263–286.
Mathematical Reviews (MathSciNet): MR1741808
Zentralblatt MATH: 1028.60087
Digital Object Identifier: doi:10.1016/S0304-4149(99)00097-6
Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Prob. 14, 378–418.
Mathematical Reviews (MathSciNet): MR2023025
Zentralblatt MATH: 1041.60024
Digital Object Identifier: doi:10.1214/aoap/1075828056
Project Euclid: euclid.aoap/1075828056
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. 154–165.
Mathematical Reviews (MathSciNet): MR1340440
Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica 29, 3–33.
Mathematical Reviews (MathSciNet): MR1887296
Digital Object Identifier: doi:10.1007/BF02679611
Volkovich, Y. and Litvak, N. (2010). Asymptotic analysis for personalized web search. Adv. Appl. Prob. 42, 577–604.
Mathematical Reviews (MathSciNet): MR2675117
Zentralblatt MATH: 1209.68180
Digital Object Identifier: doi:10.1239/aap/1275055243
Project Euclid: euclid.aap/1275055243
Waymire, E. C. and Williams, S. C. (1995). Multiplicative cascades: dimension spectra and dependence. J. Fourier Anal. Appl. 1995, 589–609.
Mathematical Reviews (MathSciNet): MR1364911
Zentralblatt MATH: 0889.60050
Advances in Applied Probability
