Martingale convergence and the stopped branching random walk

Reference:

Kyprianou, A. E., 2000. Martingale convergence and the stopped branching random walk. Probability Theory and Related Fields, 116 (3), pp. 405-419.

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Official URL:

http://dx.doi.org/10.1007/s004400050256

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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