离散蛇的从属化
Subordination of discrete snakes
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中文总结 AI 辅助
本文研究离散蛇的从属化，刻画了Bienaymé-Galton-Watson树从属树的分布，证明其仍为BGW树且稳定指数变为(α+1)/2，并得到最大位移尾部及尺度极限。
中文摘要 AI 辅助
受随机几何应用的启发，我们研究了离散情形下蛇的从属化概念。更确切地说，给定一棵由树 $T$ 索引且步长取值于 $\{...,-1,0,1\}$ 的随机游走 $W$，我们考虑其从属树，该从属树通过收缩 $T$ 中所有不导致 $W$ 新记录的边而得到。当底层树 $T$ 是 Bienaymé-Galton-Watson 树时，我们刻画了该从属树的分布。特别地，当 $T$ 具有临界后代分布且该分布属于参数 $\alpha\in(1,2]$ 的 $\alpha$-稳定吸引域，并且在步长轻尾假设下，我们证明相应的从属树本身是一棵 Bienaymé-Galton-Watson 树，其后代分布属于 $\frac{\alpha+1}{2}$-稳定吸引域。在此过程中，我们在最小假设下获得了该稳定区域内临界分支随机游走 $W$ 最大位移的渐近尾部。最后，我们利用这些结果证明了关于从属树的尺度极限命题。
英文摘要
Motivated by applications in random geometry, we investigate the notion of subordination of snakes in the discrete setup. More precisely, given a random walk $W$ indexed by a tree $T$ and with steps in $\{...,-1,0,1\}$, we consider its subordinate tree obtained by contracting every edge of $T$ that does not lead to a new record of the walk $W$. When the underlying tree $T$ is a Bienaymé-Galton-Watson tree, we characterize the distribution of this subordinate tree. In particular, when $T$ has a critical offspring distribution in an $α$-stable domain of attraction with $α\in(1,2]$, and under a light tails assumption on the steps, we prove that the associated subordinate tree is itself a Bienaymé-Galton-Watson tree with an offspring distribution in an $\frac{α+1}{2}$-stable domain of attraction. Along the way, we obtain the asymptotic tail of the maximal displacement of the critical branching random walk $W$ in this stable regime, under minimal assumptions. Finally, we use these results to prove scaling limit statements about the subordinate tree.
