For a continuous-time catalytic branching random walk (CBRW) on $${\mathbb {Z}}$$ Z , with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include regular variation of the jump distribution tail for underlying random walk and the well-known $$L\log L$$ L log L condition for the offspring numbers. In our classification, given in Bulinskaya (Theory Probab Appl 59(4):545–566, 2015), the analysis refers to supercritical CBRW. The principal result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to a non-trivial law. An explicit formula is provided for this normalization, and nonlinear integral equations are obtained to determine the limiting distribution function. The novelty consists in establishing the weak convergence for CBRW with “heavy” tails, in contrast to the known behavior in case of “light” tails of the random walk jumps. The new tools such as “many-to-few lemma” and spinal decomposition appear ineffective here. The approach developed in this paper combines the techniques of renewal theory, Laplace transform, nonlinear integral equations and large deviations theory for random sums of random variables.

中文翻译：

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尾端规律变化的催化分支随机游走的最大值

对于 $${\mathbb {Z}}$$ Z 上的连续时间催化分支随机游走 (CBRW)，使用任意有限数量的催化剂，我们研究了当时间趋于无穷大时最右边粒子位置的渐近行为. 温和的要求包括基础随机游走的跳跃分布尾部的规则变化和后代数量的众所周知的 $$L\log L$$L log L 条件。在我们的分类中，在 Bulinskaya (Theory Probab Appl 59(4):545–566, 2015) 中给出的分析是指超临界 CBRW。主要结果表明，经过适当的归一化后，CBRW 的最大值在分布中收敛于非平凡法则。为这种归一化提供了一个明确的公式，并获得非线性积分方程来确定极限分布函数。新颖之处在于为具有“重”尾巴的 CBRW 建立弱收敛，这与随机游走跳跃的“轻”尾巴情况下的已知行为形成对比。“多对少引理”和脊柱分解等新工具在这里似乎无效。本文开发的方法结合了更新理论、拉普拉斯变换、非线性积分方程和大偏差理论的技术，用于随机变量的随机和。