Abstract.
We discuss the construction of stopping lines in the branching random walk and thus the existence of a class of supermartingales indexed by sequences of stopping lines. Applying a method of Lyons (1997) and Lyons, Pemantle and Peres (1995) concerning size biased branching trees, we establish a relationship between stopping lines and certain stopping times. Consequently we develop conditions under which these supermartingales are also martingales. Further we prove a generalization of Biggins' Martingale Convergence Theorem, Biggins (1977a) within this context.
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Received: 21 December 1998 / Revised version: 28 June 1999 / Published online: 7 February 2000
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Kyprianou, A. Martingale convergence and the stopped branching random walk. Probab Theory Relat Fields 116, 405–419 (2000). https://doi.org/10.1007/s004400050256
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DOI: https://doi.org/10.1007/s004400050256
Keywords
- Peris
- Random Walk
- Convergence Theorem
- Martingale Convergence Theorem