Martingale convergence and the stopped branching random walk

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.