The supremum of a martingale related to a branching random walk
Authors:
O. Iksanov and P. Negadaĭlov
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 74 (2007), 49-57
MSC (2000):
Primary 60J80, 60E99; Secondary 60G42
DOI:
https://doi.org/10.1090/S0094-9000-07-00697-7
Published electronically:
June 29, 2007
MathSciNet review:
2336778
Full-text PDF Free Access
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Additional Information
Abstract: Let $W_n$, $n\geq 1$, be a standard martingale constructed from a supercritical branching random walk where the number of individuals in a generation is allowed to be infinite with a positive probability. We find the behavior of $\mathsf {P}\{\sup _n W_n>x\}$ as $x\to \infty$ under certain conditions.
References
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- Aleksander M. Iksanov, Elementary fixed points of the BRW smoothing transforms with infinite number of summands, Stochastic Process. Appl. 114 (2004), no. 1, 27–50. MR 2094146, DOI https://doi.org/10.1016/j.spa.2004.06.002
- Aleksander M. Iksanov and Che-Soong Kim, On a Pitman-Yor problem, Statist. Probab. Lett. 68 (2004), no. 1, 61–72. MR 2064686, DOI https://doi.org/10.1016/j.spl.2004.01.015
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References
- J. D. Biggins, Growth rates in the branching random walk, Z. Wahrscheinlichkeitstheorie verw. Gebiete 48 (1979), 17–34. MR 533003 (80e:60095)
- C. M. Goldie, Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Prob. 1 (1991), 126–166. MR 1097468 (93i:60118)
- A. K. Grincevičjus, On a limit distribution for a random walk on lines, Litovsk. Mat. Sb. 15 (1975), no. 4, 79–91. MR 0448571 (56:6877)
- O. M. Iksanov, A conjecture on the behavior of tails of fixed points of the shot noise transform, Teor. Imovirnost. Matem. Statist. 69 (2003), 37–41; English transl. in Theor. Probability Math. Statist. 69 (2004), 55–60. MR 2110904 (2005k:60051)
- A. M. Iksanov, Elementary fixed points on the BRW smoothing transforms with infinite number of summands, Stoch. Proc. Appl. 114 (2004), 27–50. MR 2094146 (2005i:60168)
- A. M. Iksanov and C. S. Kim, On a Pitman–Yor problem, Stat. Prob. Lett. 68 (2004), 61–72. MR 2064686 (2005f:60037)
- Q. Liu, On generalized multiplicative cascades, Stoch. Proc. Appl. 86 (2000), 263–286. MR 1741808 (2001b:60102)
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Additional Information
O. Iksanov
Affiliation:
Faculty for Cybernetics, National Taras Shevchenko University, Glushkov Avenue, 6, Kyiv, 03127, Ukraine
Email:
iksan@unicyb.kiev.ua
P. Negadaĭlov
Affiliation:
Faculty for Cybernetics, National Taras Shevchenko University, Glushkov Avenue, 6, Kyiv, 03127, Ukraine
Email:
npasha@ukr.net
Keywords:
Branching random walk,
supremum of a martingale,
renewal equation
Received by editor(s):
March 16, 2005
Published electronically:
June 29, 2007
Additional Notes:
The authors are indebted to O. K. Zakusylo for a careful reading of the paper and for a number of helpful comments
Article copyright:
© Copyright 2007
American Mathematical Society
