Symmetric fixed points of a smoothing transformation

Advances in Applied Probability

Volumn 35, Issue 2, 2003, Pages 377-394

  Caliebe, Amke ^a  

^a UNIVERSITY OF KIEL   (Germany)

Author keywords

Convergence of triangular schemes; Distributional fixed point equation; Functional equation; Infinitely divisible distribution; Representation of solutions; Stable distribution

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; FUNCTION EVALUATION; MATHEMATICAL MODELS; METHOD OF MOMENTS; NORMAL DISTRIBUTION; OPTIMIZATION; RANDOM PROCESSES; SET THEORY; THEOREM PROVING;

CONVERGENCE OF TRIANGULAR SCHEME; DISTRIBUTIONAL FIXED POINT EQUATION; INFINITELY DIVISIBLE DISTRIBUTIONS; REAL RANDOM VARIABLES; STABLE DISTRIBUTION; SYMMETRIC FIXED POINTS;

MATHEMATICAL TRANSFORMATIONS;

EID: 0037706919     PISSN: 00018678     EISSN: None     Source Type: Journal    
DOI: 10.1239/aap/1051201653     Document Type: Article

References (42)

1. 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.
2. Biggins, J. D. (1977a). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25-37.
3. Biggins, J. D. (1977b). Chernoff's theorem in the branching random walk. J. Appl. Prob. 14, 630-636.
4. Biggins, J. D. and Grey, D. R. (1979). Continuity of limit random variables in the branching random walk. J. Appl. Prob. 16, 740-749.
5. Biggins, J. D. and Kyprianou, A. E. (1997). Seneta-Heyde norming in the branching random walk. Ann. Prob. 25, 337-360.
6. 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/~st1jdb/.
7. 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.
8. 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.
9. Dawson, D. A. (1977). The critical measure diffusion process. Z. Wahrscheinlichkeitsth. 40, 125-145.
10. Durrett, R. and Liggett, T. (1983). Fixed points of the smoothing transformation. Z. Wahrscheinlichkeitsth. 64, 275-301.
11. Falconer, K. J. (1986). Random fractals. Math. Proc. Camb. Phil. Soc. 100, 559-582.
12. Falconer, K. J. (1987). Cut set sums and tree processes. Proc. Amer. Math. Soc. 101, 337-346.
13. Geiger, J. (2000). A new proof of Yaglom's exponential limit law. In Algorithms, Trees, Combinatorics and Probability. eds D. Gardy and A. Mokkadem, Birkhäuser, Basel, pp. 245-249.
14. Gnedenko, B. V. and Kolmogorov, A. N. (1968). Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading, MA.
15. 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.
16. Guivarc'h, Y. (1990). Sur une extension de la notion de loi semi-stable. Ann. Inst. H. Poincaré Prob. Statist. 26, 261-285.
17. Harris, T. E. (1948). Branching processes. Ann. Math. Statist. 19, 474-494.
18. Heyde, C. C. (1970). Extension of a result of Seneta for the super-critical Galton-Watson process. Ann. Math. Stat. 41, 739-742.
19. Holley, R. and Liggett, T. M. (1981). Generalized potlatch and smoothing processes. Z. Wahrscheinlichkeitsth. 55, 165-195.
20. Hutchinson, I. and Rüschendorf, L. (2000). Random fractal measures and probability metrics. Adv. Appl. Prob. 32, 925-947.
21. 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.
22. Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.
23. Kahane, J. P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, 131-145.
24. Kesten, H. and Stigum, B. P. (1966). A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Statist. 37, 1211-1223.
25. Liggett, T. M. (1978). Random invariant measures for Markov chains, and independent particle systems. Z. Wahrscheinlichkeitsth. 45, 297-313.
26. Laggett, T. M. and Spitzer, F. (1981). Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrscheinlichkeitsth. 56, 443-468.
27. Liu, Q. (1996). The exact Hausdorff dimension of a branching set. Prob. Theory Relat. Fields 104, 514-538.
28. 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.
29. Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Prob. 30, 85-112.
30. Liu, Q. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stoch. Process. Appl. 95, 83-107.
31. 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 K. B. Athreya and P. Jagers, Springer, Berlin, pp. 217-221.
32. 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.
33. Mandelbrot, B. (1974a). Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire. C. R. Acad. Sci. Paris A 278, 289-292.
34. Mandelbrot, B. (1974b). Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire: quelques extensions. C. R. Acad. Sci. Paris A 278, 355-358.
35. Maudlin, R. D. and Williams, S. C. (1986). Random constructions, asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295, 325-346.
36. Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, Berlin.
37. Rachev, S. T. and Rüschendorf, L. (1994). Propagation of chaos and contraction of stochastic mappings. Siberian Adv. Math. 1, 114-150.
38. Rösler, U. (1992). A fixed-point theorem for distributions. Stoch. Process. Appl. 42, 195-214.
39. 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/.
40. Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica 29, 3-33.
41. Rösler, U., Topchii, V. and Vatutin, V. A. (2002). Convergence rate for stable weighted branching processes. In Mathematics and Computer Sci. II, eds B. Chauvin, P. Flajolet and A. Mokkadem. Birkhäuser, Basel, pp. 441-453.
42. Stigum, B. P. (1966). A theorem on the Galton-Watson process. Ann. Math. Statist. 37, 696-698.