Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 11 (2006) > Paper 2 open journal systems 


A Stochastic Fixed Point Equation Related to Weighted Branching with Deterministic Weights

Gerold Alsmeyer, Inst. Math. Statistics, Dept. Math. and Computer Science, University of Münster
Uwe Rösler, Math. Seminar, University of Kiel


Abstract
 For real numbers $C,T_{1},T_{2},...$ we find all solutions $mu$ to the
stochastic fixed point equation $Weqdistsum_{jge 1}T_{j}W_{j}+C$,
where $W,W_{1},W_{2},...$ are independent real-valued random variables with
distribution $mu$ and $eqdist$ means equality in distribution. All solutions are
infinitely divisible. The set of solutions depends on the closed multiplicative
subgroup of ${Bbb R}_{*}={Bbb R}backslash{0}$ generated by the $T_{j}$. If this group
is continuous, i.e. ${Bbb R}_{*}$ itself or the positive halfline ${BbbR}_{+}$, then
all nontrivial fixed points are stable laws. In the remaining (discrete) cases
further periodic solutions arise. A key observation is that the L'evy measure of
any fixed point is harmonic with respect to $Lambda=sum_{jge 1}delta_{T_{j}}$,
i.e. $Gamma=GammastarLambda$, where $star$ means multiplicative convolution.
This will enable us to apply the powerful Choquet-Deny theorem. 


Full text: PDF

Pages: 27-56

Published on: January 26, 2006
Bibliography
  1. Arbeiter, Matthias. Random recursive construction of self-similar fractal measures. The noncompact case. Probab. Theory Related Fields 88 (1991), no. 4, 497--520. MR1105715 (92m:60040)
  2. Biggins, J. D. Uniform convergence of martingales in the branching random walk. Ann. Probab. 20 (1992), no. 1, 137--151. MR1143415 (93b:60094)
  3. Biggins, J. D.; Kyprianou, A. E. Seneta-Heyde norming in the branching random walk. Ann. Probab. 25 (1997), no. 1, 337--360. MR1428512 (98a:60118)
  4. Burton, Robert M.; Rösler, Uwe . An $Lsb 2$ convergence theorem for random affine mappings. J. Appl. Probab. 32 (1995), no. 1, 183--192. MR1316801 (96b:60011)
  5. Caliebe, Amke. Symmetric fixed points of a smoothing transformation. Adv. in Appl. Probab. 35 (2003), no. 2, 377--394. MR1970480 (2004f:60036)
  6. Choquet, Gustave; Deny, Jacques. Sur l'équation de convolution $µ=µ*sigma $. (French) C. R. Acad. Sci. Paris 250 1960 799--801. MR0119041 (22 #9808)
  7. Chow, Yuan Shih; Teicher, Henry. Probability theory. Independence, interchangeability, martingales. Third edition. Springer Texts in Statistics. Springer-Verlag, New York, 1997. xxii+488 pp. ISBN: 0-387-98228-0 MR1476912 (98e:60003)
  8. Davies, Laurie. A theorem of Deny with applications to characterization problems. Analytical methods in probability theory (Oberwolfach, 1980), pp. 35--41, Lecture Notes in Math., 861, Springer, Berlin-New York, 1981. MR0655258 (84h:62024)
  9. Davies, Laurie; Shimizu, Ryoichi. On identically distributed linear statistics. Ann. Inst. Statist. Math. 28 (1976), no. 3, 469--489. MR0431467 (55 #4465)
  10. Dufresne, Daniel. The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuar. J. 1990, no. 1-2, 39--79. MR1129194 (92i:62195)
  11. Durrett, Richard; Liggett, Thomas M. Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 275--301. MR0716487 (85e:60059)
  12. Embrechts, Paul; Goldie, Charles M. Perpetuities and random equations. Asymptotic statistics (Prague, 1993), 75--86, Contrib. Statist., Physica, Heidelberg, 1994. MR1311930 (95m:60093)
  13. Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp. MR0270403 (42 #5292)
  14. Gnedenko, B. W.; Kolmogorov, A. N. Grenzverteilungen von Summen unabhängiger Zufallsgrössen. (German) Wissenschaftliche Bearbeitung der deutschen Ausgabe: Prof. Dr. Josef Heinhold. Mathematische Lehrbücher und Monographien, II. Abt., Bd. IX Akademie-Verlag, Berlin 1959 viii+279 pp. MR0107295 (21 #6020)
  15. Graf, Siegfried. Statistically self-similar fractals. Probab. Theory Related Fields 74 (1987), no. 3, 357--392. MR0873885 (88c:60038)
  16. Guivarc'h, Yves. Sur une extension de la notion de loi semi-stable. (French) [On an extension of the notion of semistable law] Ann. Inst. H. Poincaré Probab. Statist. 26 (1990), no. 2, 261--285. MR1063751 (91i:60141)
  17. Johansen, S. An application of extreme point methods to the representation of infinitely divisible distributions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5 1966 304--316. MR0203765 (34 #3613)
  18. Kahane, J.-P.; Peyrière, J. Sur certaines martingales de Benoit Mandelbrot. Advances in Math. 22 (1976), no. 2, 131--145. MR0431355 (55 #4355)
  19. Kesten, Harry. Random difference equations and renewal theory for products of random matrices. Acta Math. 131 (1973), 207--248. MR0440724 (55 #13595)
  20. Khintchine, A. Zur Theorie der unbeschränkt teilbaren Verteilungsgesetze. (German) {it Rec. Math. Moscou, n. Ser.} 2 (1937), 79-117. Zbl0016.4100
  21. Liu, Quansheng. Sur une équation fonctionnelle et ses applications: une extension du théorème de Kesten-Stigum concernant des processus de branchement. (French) [On a functional equation and its applications: an extension of the Kesten-Stigum theorem on branching processes] Adv. in Appl. Probab. 29 (1997), no. 2, 353--373. MR1450934 (98j:60124)
  22. Liu, Quansheng. Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. in Appl. Probab. 30 (1998), no. 1, 85--112. MR1618888 (99f:60151)
  23. Mandelbrot, Benoit. Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire. C. R. Acad. Sci. Paris Sér. A 278 (1974), 289--292. MR0431351 (55 #4352a)
  24. Rachev, S. T.; Rüschendorf, L. Probability metrics and recursive algorithms. Adv. in Appl. Probab. 27 (1995), no. 3, 770--799. MR1341885 (96m:60005)
  25. Ramachandran, B.; Lau, Ka-Sing; Gu, Hua Min. On characteristic functions satisfying a functional equation and related classes of simultaneous integral equations. Sankhy=a Ser. A 50 (1988), no. 2, 190--198. MR1056443 (91g:60023)
  26. Rösler, Uwe. A limit theorem for "Quicksort". RAIRO Inform. Théor. Appl. 25 (1991), no. 1, 85--100. MR1104413 (92f:68028)
  27. Rösler, Uwe. A fixed point theorem for distributions. Stochastic Process. Appl. 42 (1992), no. 2, 195--214. MR1176497 (93k:60038)
  28. Rösler, Uwe. The weighted branching process. Dynamics of complex and irregular systems (Bielefeld, 1991), 154--165, Bielefeld Encount. Math. Phys., VIII, World Sci. Publishing, River Edge, NJ, 1993. MR1340440 (96f:60148)
  29. Rösler, Uwe. A fixed point equation for distributions. Report No. 98-7, Univ. Kiel (1998). Math. Review number not available.
  30. Rösler, U.; Rüschendorf, L. The contraction method for recursive algorithms. Average-case analysis of algorithms (Princeton, NJ, 1998). Algorithmica 29 (2001), no. 1-2, 3--33. MR1887296 (2003c:68096)
  31. Shimizu, Ryoichi. Solution to a functional equation and its application to some characterization problems. Sankhy=a Ser. A 40 (1978), no. 4, 319--332. MR0589287 (81j:60021)













Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489