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Fixed points of a generalized smoothing transformation and applications to the branching random walk

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Quansheng Liu
Quansheng Liu*
Affiliation:
Université de Rennes 1
*
∗ Postal address: Institut Mathématique de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France.
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Abstract

Let {Ai : i ≥ 1} be a sequence of non-negative random variables and let M be the class of all probability measures on [0,∞]. Define a transformation T on M by letting Tμ be the distribution of ∑i=1∞AiZi, where the Zi are independent random variables with distribution μ, which are also independent of {Ai}. Under first moment assumptions imposed on {Ai}, we determine exactly when T has a non-trivial fixed point (of finite or infinite mean) and we prove that all fixed points have regular variation properties; under moment assumptions of order 1 + ε, ε > 0, we find all the fixed points and we prove that all non-trivial fixed points have stable-like tails. Convergence theorems are given to ensure that each non-trivial fixed point can be obtained as a limit of iterations (by T) with an appropriate initial distribution; convergence to the trivial fixed points δ0 and δ∞ is also examined, and a result like the Kesten-Stigum theorem is established in the case where the initial distribution has the same tails as a stable law. The problem of convergence with an arbitrary initial distribution is also considered when there is no non-trivial fixed point. Our investigation has applications in the study of: (a) branching processes; (b) invariant measures of some infinite particle systems; (c) the model for turbulence of Yaglom and Mandelbrot; (d) flows in networks and Hausdorff measures in random constructions; and (e) the sorting algorithm Quicksort. In particular, it turns out that the basic functional equation in the branching random walk always has a non-trivial solution.

Keywords

Smoothing transformationbranching processesbranching random walksMandelbrot's martingalefunctional equationstable law

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 85 - 112
Copyright
Copyright © Applied Probability Trust 1998 

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References

Athreya, K.B. (1971). A note on a functional equation arising in Galton–Watson branching processes. J. Appl. Prob. 8, 589–598.CrossRefGoogle Scholar
Athreya, K.B. and Ney, P.E. (1972). Branching Processes. Springer, Berlin.Google Scholar
Ben Nasr, F. (1987). Mesures aléatoires de Mandelbrot associés à des substitutions. C. R. Acad. Sci. Paris, 304, 255–258.Google Scholar
Biggins, J.D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37.Google Scholar
Biggins, J.D. and Kyprianou, A.E. (1996). Branching random walk: Seneta–Heyde norming. Trees: Progress in probability 40 31–50, eds. Chauvin, B., Cohen, S., Rouault., A Birkhäuser, Boston.Google Scholar
Biggins, J.D. and Kyprianou, A.E. (1997). Seneta-Heyde norming in the branching random walk. Ann. Prob. 25, 337–360.Google Scholar
Bingham, N.H. and Doney, R.A. (1974). Asymptotic properties of supercritical branching processes I: The Galton–Watson process. Adv. Appl. Prob. 6, 711–731.Google Scholar
Bingham, N.H. and Doney, R.A. (1975). Asymptotic properties of supercritical branching processes II: Crump–Mode and Jirina processes. Adv. Appl. Prob. 7, 66–82.CrossRefGoogle Scholar
Bingham, N.H., Goldie, C.M. and Teugels, J.D. (1987). Regular Variation. Cambridge University Press, Cambridge.Google Scholar
Chauvin, B. and Rouault, A. (1996). Boltzmann-Gibbs weights in the branching random walk. In Classical and modern branching processes, IMA Proceedings 84, 41–50 ed. Athreya, K.B. and Jagers, P., Springer, New York.Google Scholar
Collet, P. and Koukiou, F. (1992). Large deviations for multiplicative chaos. Commun. Math. Phys. 147, 329–342.CrossRefGoogle Scholar
Cohn, H. (1982). Norming constants for the finite mean supercritical Bellman–Harris processes. Z. Wahrscheinlichkeitsth. 61, 189–205.CrossRefGoogle Scholar
Cohn, H (1985). A martingale approach to supercritical (CMJ) branching processes. Ann. Prob. 13, 1179–1191.Google Scholar
Crump, K. and Mode, C.J. (1968). A general age-dependent branching process (I). J. Math. Anal. Appl. 24, 497–508.Google Scholar
Crump, K. and Mode, C.J. (1969). A general age-dependent branching process (II). J. Math. Anal. Appl. 25, 8–17.CrossRefGoogle Scholar
Doney, R.A. (1972). A limit theorem for a class of supercritical branching processes. J. Appl. Prob. 9, 707–724.Google Scholar
Doney, R.A. (1973). On a functional equation for general branching processes. J. Appl. Prob. 10, 497–508.Google Scholar
Durrett, R. and Liggett, T. (1983). Fixed points of the smoothing transformation. Z. Wahrscheinlichkeitsth. 64, 275–301.CrossRefGoogle Scholar
Falconer, K.J. (1986). Random fractals. Math. Proc. Camb. Phil. Soc. 100, 559–582.Google Scholar
Falconer, K.J. (1987). Cut-set processes and tree processes. Proc. Amer. Math. Soc. 101, 337–346.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications. Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Fortet, R., Mourier, E (1953). Convergence de la répartition empirique vers la répartition théorique. Ann. Sci. Ecole Normale Sup. Série 3, 70, 266–285.Google Scholar
Franchi, J. (1993). Chaos multiplicatif: un traitement simple et complet de la fonction de la partition. Prépublication no. 148 du Laboratoire de Probabilités de l'Université Paris VI.Google Scholar
Guivarc'h, Y., (1990). Sur une extension de la notion semi-stable. Ann. Inst. Henri Poincaré 26, 261–285.Google Scholar
Harris, T.E. (1948). Branching processes. Ann. Math. Statist. 19, 474–494.Google Scholar
Heyde, C.C. (1970). Extension of a result of Seneta for the supercritical Galton–Watson process. Ann. Math. Statist. 41, 739–742.Google Scholar
Holley, R. and Liggett, T. (1981). Generalized potlatch and smoothing processes. Z. Wahrscheinlichkeitsth. 55, 165–195.CrossRefGoogle Scholar
Holley, R. and Waymire, E.C. (1992). Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl. Prob. 2, 819–845.Google Scholar
Kahane, J.P. (1987). Multiplications aléatoires et dimensions de Hausdorff. Ann. Inst. Henri Poincaré 23, 289–296.Google Scholar
Kahane, J.P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, 131–145.Google Scholar
Kesten, H. and Stigum, B.P. (1966). A limit theorem for multidimensional Galton–Watson processes, Ann. Math. Statist. 37, 1211–1223.Google Scholar
Liu, Q. (1993). Sur quelques problèmes à propos des processus de branchement, des flots dans les réseaux et des mesures de Hausdorff associées. Thèse, Université Paris.Google Scholar
Liu, Q. (1996a). The exact Hausdorff dimennsion of a branching set. Prob. Theory Rel. Fields 104, 515–538.Google Scholar
Liu, Q. (1996b). The growth of an entire characteristic function and the tail probability of the limit of a tree martingale. In Trees: Progress in probability 40, 51–80, eds Chauvin, B., Cohen, S., Rouault., A Birkhäuser, Boston.Google Scholar
Liu, Q. (1997). Sur une équation fonctionnelle et ses applications: une extension du théorème de Kesten–Stigum concernant des processus de branchement. Adv. Appl. Prob. (To appear.).Google Scholar
Mandelbrot, B. (1974a). Multiplications aléatoires et distributions invariantes par moyenne pondérée aléatoire. C. R. Acad. Sci. Paris, Série A 278, 325–346.Google Scholar
Mandelbrot, B. (1974b). Multiplications aléatoires et distributions invariantes par moyenne pondérée aléatoire: quelques extensions. C. R. Acad. Sci. Paris, Série A 278, 355–358.Google Scholar
Mauldin, R.A. and Williams, S.C. (1986). Random constructions, asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295, 325–346.Google Scholar
Rösler (1992). A fixed point theorem for distributions. Stoch. Proc. Appl. 42, 195–214.CrossRefGoogle Scholar
Royer, G. (1984). Distance de Fortet–Mourier et fonctions log-concaves. Ann. Sci. de Clermont-Ferrand 78.Google Scholar
Seneta, E. (1968). On recent theorems concerning the supercritical Galton–Watson process. Ann. Math. Statist. 39, 2098–2102.CrossRefGoogle Scholar
Seneta, E. (1969). Functional equations and the Galton–Watson process. Adv. Appl. Prob. 1, 1–42.CrossRefGoogle Scholar
Seneta, E. (1974). Regular varying functions in the theory of simple branching processes. Adv. Appl. Prob. 6, 408–420.Google Scholar
Waymire, E.C. and Williams, S.C. (1995). Multiplicative cascades: Dimension spectra and dependence. J. Fourier Anal. Appl. (Kahane Special Issue), 589–609.Google Scholar