具有显式收敛速度的Schröder分支过程的左尾展开
Left-tail expansions for Schröder branching processes with explicit convergence rates
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中文总结 AI 辅助
该研究移除了Schröder分支过程鞅极限密度左尾展开证明中的两个假设，推导了带显式收敛速度的展开系数界，分析了临界角行为及其对振荡修正项的影响。
中文摘要 AI 辅助
在先前的研究中，Schröder分支过程中鞅极限的密度被表示为带有振荡项的收敛双幂级数，该证明依赖于两个假设，其中一个假设对后代生成函数的Julia集在1附近的临界角施加了几何限制。在本文中，我们证明这两个假设均可被移除。我们推导了展开系数的显式界，这意味着该双级数的局部一致收敛性；系数在一个求和指标上指数衰减，其衰减速率由临界角显式确定，在另一个求和指标上则超指数衰减。最后，我们研究了临界角的行为并描述其趋近最小值的区域，该分析揭示了Julia集的几何结构如何影响左尾展开中振荡修正项的幅度。
英文摘要
In previous work, the density of the martingale limit in Schröder branching processes was expressed as a convergent double power-law series with oscillatory terms. The proof relied on two assumptions, one of which imposed a geometric restriction on the critical angle of the Julia set of the offspring generating function near $1$. In this paper, we show that both assumptions can be removed. We derive explicit bounds on the expansion coefficients, which imply locally uniform convergence of the double series. The coefficients decay exponentially in one summation index, with the rate explicitly determined by the critical angle, and super-exponentially in the other. Finally, we investigate the behavior of the critical angle and describe regimes in which it approaches its minimum. This analysis shows how the geometry of the Julia set influences the magnitude of the oscillatory corrections in the left-tail expansion.
