论文标题
大偏差，最大的分支随机步行，带有拉伸指数尾巴
Large deviations for the maximum of a branching random walk with stretched exponential tails
论文作者
论文摘要
在不满足Cramér的情况的情况下，我们在一维分支随机步行中证明了最右边粒子的位置的较大偏差结果，并用$ m_n $表示。更确切地说，我们考虑具有拉伸指数上和下尾的步长分布，即两个尾巴衰变为$ e^{ - | t |^r} $，对于某些$ r \ in（0,1）$。众所周知，在这种情况下，$ m_n $成长为$ n^{1/r} $，尤其比$ n $中的线性更快。我们的主要结果是$ n^{ - 1/r} m_n $定律的大偏差原则。在证明中，我们使用的比较与（随机数的）独立随机步行的最大比较，以$ \ tilde m_n $表示，并且我们还显示了$ n^{ - 1/r} \ tilde m_n $的法律的较大偏差原理。
We prove large deviation results for the position of the rightmost particle, denoted by $M_n$, in a one-dimensional branching random walk in a case when Cramér's condition is not satisfied. More precisely we consider step size distributions with stretched exponential upper and lower tails, i.e.~both tails decay as $e^{-|t|^r}$ for some $r\in( 0,1)$. It is known that in this case, $M_n$ grows as $n^{1/r}$ and in particular faster than linearly in $n$. Our main result is a large deviation principle for the laws of $n^{-1/r}M_n$ . In the proof we use a comparison with the maximum of (a random number of) independent random walks, denoted by $\tilde M_n$, and we show a large deviation principle for the laws of $n^{-1/r}\tilde M_n$ as well.
