We consider the boundary case in a one-dimensional supercritical branching random walk, and study two of the most important martingales: the additive martingale (Wn) and the derivative martingale (Dn). It is known that upon the system's survival, Dn has a positive almost sure limit (Biggins and Kyprianou [9]), whereas Wn converges almost surely to 0 (Lyons [22]). Our main result says that after a suitable normalization, the ratio Wn/Dn converges in probability, upon the system's survival, to a positive constant.

Original language | English |
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Place of Publication | Eindhoven |
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Publisher | Eurandom |
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Number of pages | 38 |
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Publication status | Published - 2011 |
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Name | Report Eurandom |
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Volume | 2011016 |
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ISSN (Print) | 1389-2355 |
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