Fluctuations of Biggins’ martingales at complex parameters
Fluctuations of Biggins’ martingales at complex parameters
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DOI：
10.1214/20-aihp1046
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发表时间：
2018-06
期刊：
影响因子：
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通讯作者：
A. Iksanov;Konrad Kolesko;M. Meiners
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文献类型：
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作者：
A. Iksanov;Konrad Kolesko;M. Meiners
超临界分支随机游走的长期行为可以用实数或复数参数化的Biggins鞅来描述和分析。对这些复参数鞅的研究是一个相当新的课题。假设鞅收敛于非简并极限的某些充分条件成立，我们研究了鞅在其极限附近的涨落。我们发现了三种不同的体制。首先，我们证明了对于绝对值较小的参数，波动是高斯的，极限律是实标准正态律或复标准正态律的尺度混合。我们还讨论了这个阶段的边界。其次，我们在参数空间中找到一个区域，其中的鞅波动由分支随机游走中的最小位置决定。最后，存在一个临界区域（通常在参数集的边界上，其鞅收敛于非退化极限），其中波动是类稳定的，极限律是满足类似于稳定性的不变性的随机停止的L\'evy过程的定律。
The long-term behavior of a supercritical branching random walk can be described and analyzed with the help of Biggins' martingales, parametrized by real or complex numbers. The study of these martingales with complex parameters is a rather recent topic. Assuming that certain sufficient conditions for the convergence of the martingales to non-degenerate limits hold, we investigate the fluctuations of the martingales around their limits. We discover three different regimes. First, we show that for parameters with small absolute values, the fluctuations are Gaussian and the limit laws are scale mixtures of the real or complex standard normal law. We also cover the boundary of this phase. Second, we find a region in the parameter space in which the martingale fluctuations are determined by the minimal positions in the branching random walk. Finally, there is a critical region (typically on the boundary of the set of parameters for which the martingales converge to a non-degenerate limit) where the fluctuations are stable-like and the limit laws are the laws of randomly stopped L\'evy processes satisfying invariance properties similar to stability.