Convergence of complex martingales in the branching random walk: the boundary
Convergence of complex martingales in the branching random walk: the boundary
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DOI：
10.1214/17-ecp50
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发表时间：
2016-11
影响因子：
0.5
通讯作者：
Konrad Kolesko;M. Meiners
中科院分区：
文献类型：
--
作者：
Konrad Kolesko;M. Meiners
分支随机游动中鞅的一致收敛。可能是安妮20（1）：137--151，1992]证明了开集$\Lambda \subseteq \mathbb{C}^d$中的$d$维超临界分支随机游动中的可加鞅在复参数$\lambda$下的局部一致收敛性。我们研究了参数从$\Lambda$的边界$\partial \Lambda $所对应的鞅。边界可以分解为几个部分。边界上可能有一部分不存在鞅，在另一部分存在鞅，但在极限上发散或消失。在其余部分，有收敛到一个非退化极限。给出这种收敛性的论点也适用于$\Lambda$，并且需要比Biggins使用的更弱的矩假设。
Biggins [Uniform convergence of martingales in the branching random walk. {\em 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. There may be a part of the boundary, on which the martingales do not exist, on other parts it exists, but diverges or vanishes in the limit. In the remaining part, there is convergence to a non-degenerate limit. The arguments that give this convergence also apply in $\Lambda$ and require weaker moment assumptions than the ones used by Biggins.