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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

 

The supremum of a martingale related to a branching random walk


Authors: O. Iksanov and P. Negadailov
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 74 (2006).
Journal: Theor. Probability and Math. Statist. 74 (2007), 49-57
MSC (2000): Primary 60J80, 60E99; Secondary 60G42
Published electronically: June 29, 2007
MathSciNet review: 2336778
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Abstract | References | Similar Articles | 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.


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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. Negadailov
Affiliation: Faculty for Cybernetics, National Taras Shevchenko University, Glushkov Avenue, 6, Kyiv, 03127, Ukraine
Email: npasha@ukr.net

DOI: http://dx.doi.org/10.1090/S0094-9000-07-00697-7
PII: S 0094-9000(07)00697-7
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