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- W2550881870 abstract "Biggins [Uniform convergence of martingales in the branching random walk. Ann. Probab., 20(1):137–151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $lambda $ from an open set $Lambda subseteq mathbb{C} ^d$. We investigate the martingales corresponding to parameters from the boundary $partial Lambda $ of $Lambda $. The boundary can be decomposed into several parts. We demonstrate by means of an example that there may be a part of the boundary, on which the martingales do not exist. Where the martingales exist, they may diverge, vanish in the limit or converge to a non-degenerate limit. We provide mild sufficient conditions for each of these three types of limiting behaviors to occur. The arguments that give convergence to a non-degenerate limit also apply in $Lambda $ and require weaker moment assumptions than the ones used by Biggins." @default.
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- W2550881870 date "2017-01-01" @default.
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- W2550881870 title "Convergence of complex martingales in the branching random walk: the boundary" @default.
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- W2550881870 doi "https://doi.org/10.1214/17-ecp50" @default.
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