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Large deviations and almost sure convergence for the extremes of branching Lévy processes
Runjia Luo, Yan-Xia Ren, Renming Song, Rui Zhang · 2025-09-30 · via math.PR updates on arXiv.org

In this paper, we investigate the asymptotic behavior of supercritical branching Markov processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. Recently, Ren et al. [Appl. Probab. 61 (2024)] studied the weak convergence of the extremes of $\{\mathbb{X}_t, t \ge0\}$. In this paper, we establish the large deviation of $\{\mathbb{X}_t, t \ge0\}$ as well as some almost sure convergence results of the maximum of $\mathbb{X}_t$.

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