论文标题
带有拉伸指数尾巴的分支随机步行的最大步行
The maximum of a branching random walk with stretched exponential tails
论文作者
论文摘要
在台阶尺寸分布具有拉伸指数尾巴，尤其是没有有限的指数矩时，我们研究了一维的分支随机步行。步长$ x $的尾巴衰减为$ \ mathbb {p} [x \ geq t] \ sim a \ exp（-λt^r）对于某些常数$ a，λ> 0 $，其中$ r \ in（0,1）$。我们详细描述了最右边粒子位置的渐近行为，证明了几乎纯净的限制定理，法律收敛和一些积分测试。限制定理揭示了有趣的差异，这两个制度$ r \ in（0，2/3）$和$ r \ in（2/3，1）$，在边界案例$ r = 2/3 $中却有不同的限制。
We study the one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size $X$ decays as $\mathbb{P}[X \geq t] \sim a \exp(-λt^r)$ for some constants $a, λ> 0$ where $r \in (0,1)$. We give a detailed description of the asymptotic behaviour of the position of the rightmost particle, proving almost-sure limit theorems, convergence in law and some integral tests. The limit theorems reveal interesting differences betweens the two regimes $ r \in (0, 2/3)$ and $ r \in (2/3, 1)$, with yet different limits in the boundary case $r = 2/3$.
