Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T11:12:23.489Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetric fixed points of a smoothing transformation

Part of: Limit theorems Markov processes Distribution theory - Probability

Published online by Cambridge University Press:  22 February 2016

Amke Caliebe
Amke Caliebe*
Affiliation:
Christian-Albrechts-Universität zu Kiel
*
∗ Postal address: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany. Email address: caliebe@math.uni-kiel.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Let T = (T1, T2,…) be a sequence of real random variables with ∑j=1∞1|Tj|>0 < ∞ almost surely. We consider the following equation for distributions μ: W ≅ ∑j=1∞TjWj, where W, W1, W2,… have distribution μ and T, W1, W2,… are independent. We show that the representation of general solutions is a mixture of certain infinitely divisible distributions. This result can be applied to investigate the existence of symmetric solutions for Tj ≥ 0: essentially under the condition that E ∑j=1∞Tj2 log+Tj2 < ∞, the existence of nontrivial symmetric solutions is exactly determined, revealing a connection with the existence of positive solutions of a related fixed-point equation. Furthermore, we derive results about a special class of canonical symmetric solutions including statements about Lebesgue density and moments.

Keywords

Distributional fixed-point equationfunctional equationconvergence of triangular schemesrepresentation of solutionsinfinitely divisible distributionstable distribution

MSC classification

Primary: 60E10: Characteristic functions; other transforms 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 377 - 394
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported by the German Science Foundation (DFG) grant RO 498/4-1.

References

Athreya, K. B. (1971). On the absolute continuity of the limit random variable in the supercritical Galton–Watson branching process. Proc. Amer. Math. Soc. 30, 563–565.Google Scholar
Biggins, J. D. (1977a). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37.CrossRefGoogle Scholar
Biggins, J. D. (1977b). Chernoff's theorem in the branching random walk. J. Appl. Prob. 14, 630–636.Google Scholar
Biggins, J. D. and Grey, D. R. (1979). Continuity of limit random variables in the branching random walk. J. Appl. Prob. 16, 740–749.Google Scholar
Biggins, J. D. and Kyprianou, A. E. (1997). Seneta–Heyde norming in the branching random walk. Ann. Prob. 25, 337–360.CrossRefGoogle Scholar
Biggins, J. D. and Kyprianou, A. E. (2003). The smoothing transform: the boundary case. Preprint, Department of Probability and Statistics, University of Sheffield. Available at http://www.shef.ac.uk/simst1jdb/.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
Dawson, D. A. (1977). The critical measure diffusion process. Z. Wahrscheinlichkeitsth. 40, 125–145.CrossRefGoogle 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 sums and tree processes. Proc. Amer. Math. Soc. 101, 337–346.CrossRefGoogle Scholar
Geiger, J. (2000). A new proof of Yaglom's exponential limit law. In Algorithms, Trees, Combinatorics and Probability, eds Gardy, D. and Mokkadem, A., Birkhäuser, Basel, pp. 245–249.Google Scholar
Gnedenko, B. V. and Kolmogorov, A. N. (1968). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, MA.Google Scholar
Graf, S., Mauldin, R. D. and Williams, S. C. (1988). The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc. 71, No. 381.Google Scholar
Guivarc'h, Y., (1990). Sur une extension de la notion de loi semi-stable. Ann. Inst. H. Poincaré Prob. Statist. 26, 261–285.Google Scholar
Harris, T. E. (1948). Branching processes. Ann. Math. Statist. 19, 474–494.CrossRefGoogle Scholar
Heyde, C. C. (1970). Extension of a result of Seneta for the super-critical Galton–Watson process. Ann. Math. Stat. 41, 739–742.Google Scholar
Holley, R. and Liggett, T. M. (1981). Generalized potlatch and smoothing processes. Z. Wahrscheinlichkeitsth. 55, 165–195.Google Scholar
Hutchinson, I. and Rüschendorf, L. (2000). Random fractal measures and probability metrics. Adv. Appl. Prob. 32, 925–947.Google Scholar
Hwang, H.-K. and Neininger, R. (2002). Phase change of limit laws in the quicksort recurrence under varying toll functions. SIAM J. Comput. 6, 1687–1722.CrossRefGoogle Scholar
Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.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.CrossRefGoogle Scholar
Liggett, T. M. (1978). Random invariant measures for Markov chains, and independent particle systems. Z. Wahrscheinlichkeitsth. 45, 297–313.CrossRefGoogle Scholar
Liggett, T. M. and Spitzer, F. (1981). Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrscheinlichkeitsth. 56, 443–468.Google Scholar
Liu, Q. (1996). The exact Hausdorff dimension of a branching set. Prob. Theory Relat. Fields 104, 514–538.CrossRefGoogle 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. 29, 353–373.CrossRefGoogle Scholar
Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Prob. 30, 85–112.CrossRefGoogle Scholar
Liu, Q. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stoch. Process. Appl. 95, 83–107.CrossRefGoogle Scholar
Lyons, R. (1997). A simple path to Biggins' martingale convergence for branching random walk. In Classical and Modern Branching Processes (IMA Vols Math. Appl. 84), eds Athreya, K. B. and Jagers, P., Springer, Berlin, pp. 217–221.Google Scholar
Lyons, R., Pemantel, R. and Peres, Y. (1995). Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Prob. 23, 1125–1138.Google Scholar
Mandelbrot, B. (1974a). Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire. C. R. Acad. Sci. Paris A 278, 289–292.Google Scholar
Mandelbrot, B. (1974b). Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire: quelques extensions. C. R. Acad. Sci. Paris A 278, 355–358.Google Scholar
Maudlin, R. D. and Williams, S. C. (1986). Random constructions, asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295, 325–346.Google Scholar
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, Berlin.Google Scholar
Rachev, S. T. and Rüschendorf, L. (1994). Propagation of chaos and contraction of stochastic mappings. Siberian Adv. Math. 1, 114–150.Google Scholar
Rösler, U., (1992). A fixed-point theorem for distributions. Stoch. Process. Appl. 42, 195–214.Google Scholar
Rösler, U., (1998). A fixed-point equation for distributions. Preprint, Berichtsreihe des Mathematischen Seminars Kiel, Christian-Albrechts-Universität zu Kiel. Available at http://www.numerik.uni-kiel.de/reports/.Google Scholar
Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica 29, 3–33.Google Scholar
Rösler, U., Topchii, V. and Vatutin, V. A. (2002). Convergence rate for stable weighted branching processes. In Mathematics and Computer Sci. II, eds Chauvin, B., Flajolet, P. and Mokkadem, A. Birkhäuser, Basel, pp. 441–453.Google Scholar
Stigum, B. P. (1966). A theorem on the Galton–Watson process. Ann. Math. Statist. 37, 696–698.Google Scholar