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 Electronic Journal of Probability > Vol. 10 (2005) > Paper 17 open journal systems 


Fixed Points of the Smoothing Transform: the Boundary Case

John D Biggins, University of Sheffield
Andreas E Kyprianou, Heriot-Watt University


Abstract
Let $A=(A_1,A_2,A_3,ldots)$ be a random sequence of non-negative numbers that are ultimately zero with $E[sum A_i]=1$ and $E left[sum A_{i} log A_i right] leq 0$. The uniqueness of the non-negative fixed points of the associated smoothing transform is considered. These fixed points are solutions to the functional equation $Phi(psi)= E left[ prod_{i} Phi(psi A_i) right], $ where $Phi$ is the Laplace transform of a non-negative random variable. The study complements, and extends, existing results on the case when $Eleft[sum A_{i} log A_i right]<0$. New results on the asymptotic behaviour of the solutions near zero in the boundary case, where $Eleft[sum A_{i} log A_i right]=0$, are obtained.


Full text: PDF

Pages: 609-631

Published on: June 13, 2005
Bibliography
  1. Asmussen, S. and Hering, H. (1983). Branching Processes. Birkhäuser, Boston. Math. Review 85b:60076
  2. Biggins, J.D. (1977). Martingale convergence in the branching random walk. J. Appl. Probab. 14, 25-37. Math. Review 55 #6592
  3. Biggins, J.D. (1998) Lindley-type equations in the branching random walk. Stoch. Proc. Appl 75, 105-133. Math. Review 99e:60186
  4. Biggins, J.D. and Kyprianou, A.E. (1997) Seneta-Heyde norming in the branching random walk. Ann. Probab. 25 337-360. Math. Review 98a:60118
  5. Biggins, J.D. and Kyprianou, A.E. (2004). Measure change in multitype branching. Adv. Appl. Probab. 36 544-581. Math. Review 2005f:60179
  6. Bingham, N. and Doney, R.A. (1975) Asymptotic properties of supercritical branching processes. II: Crump-Mode and Jirana processes. Adv. Appl. Probab. 7, 66-82. Math. Review 51 #14294
  7. Bramson, M. (1979), Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31, 531-581. Math. Review 58 #13382
  8. Bramson, M. (1983), Convergence of solutions of the Kolmogorov non-linear diffusion equations to travelling waves. Mem. Amer. Math. Soc. 44(285). Math. Review 84m:60098
  9. Caliebe, A. (2003) Symmetric fixed points of a smoothing transformation. Adv. Appl. Probab. 35, 377-394. Math. Review 2004f:60036
  10. Caliebe, A. and Rösler, U. (2003a) Fixed points with finite variance of a smoothing transform. Stoc. Proc. Appl. 107, 105-129. Math. Review 2004d:60039
  11. Durrett, R. and Liggett, M. (1983). Fixed points of the smoothing transform. Z. Wahrsch. verw. Gebiete, 64 , 275-301. Math. Review 85e:60059
  12. Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol.II, Wiley, New York. Math. Review 42 #5292
  13. Gatzouras, D. (2000). On the lattice case of an almost-sure renewal theorem for branching random walks. Adv. Appl. Probab. 32, 720-737. Math. Review 2001k:60118
  14. Harris, S.C. (1999) Travelling waves for the FKPP equation via probabilistic arguments, Proc. Roy. Soc. Edin. 129A 503-517. Math. Review 2000g:35109
  15. Iksanov, A.M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoc. Proc. Appl. 114, 27-50 Math. Review 2094146
  16. Iksanov, A.M. and Jurek, Z.J. (2002). On fixed points of Poisson shot noise transforms. Adv. Appl. Probab. 34, 798-825 Math. Review 2003i:60021
  17. Kahane, J.P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22 , 131-145. Math. Review 55 #4355
  18. Kyprianou, A.E. (1998) Slow variation and uniqueness of solutions to the functional equation in the branching random walk. J. Appl. Probab. 35 795-802 . Math. Review 2000d:60138
  19. Kyprianou, A.E. (2004) Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis. Ann. Inst. H. Poincaré Prob. Statist, 40, 53-72. Math. Review 2005a:60135
  20. Liu, Q. (1998) Fixed points of a generalized smoothing transform and applications to the branching processes. Adv. Appl. Probab. 30, 85-112. Math. Review 99f:60151
  21. Liu, Q. (2000) On generalized multiplicative cascades. Stoc. Proc. Appl. 86, 263-286. Math. Review 2001b:60102
  22. Lyons, R. (1997). A simple path to Biggins' martingale convergence. In Classical and Modern Branching Processes (K.B. Athreya, P. Jagers, eds.). IMA Volumes in Mathematics and its Applications 84, 217-222. Springer-Verlag, New York. Math. Review 1601749
  23. McKean, H.P. (1975) Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math. 28, 323-331. Math. Review 53 #4262 (Correction: (1976) 29, 553-554). Math. Review 54 #11534
  24. Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching process. Z. Wahrsch. verw. Gebiete. 57, 365-395. Math. Review 82m:60104
  25. Neveu, J. (1988). Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes, 1987, eds: E. Çinlar, K.L. Chung, R.K. Getoor. Progress in Probability and Statistics, 15, 223-241. Birkhäuser, Boston. Math. Review 91f:60144
  26. Pakes, A.G. (1992). On characterizations via mixed sums. Austral. J. Statist. 34, 323-339. Math. Review 93k:60043
  27. Rösler, U. (1992). A fixed point theorem for distributions. Stoc. Proc. Appl. 42, 195-214 Math. Review 93k:60038
















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Electronic Journal of Probability. ISSN: 1083-6489