论文标题
在非典型随机环境中移民的分支过程
Branching processes with immigration in atypical random environment
论文作者
论文摘要
由Kesten等人的开创性论文动机。 （1975）我们考虑了具有I.I.D.几何后代分布的分支过程。随机环境参数$ a_n $，$ n \ ge 1 $和大小-1移民的移民。与上述纸张相反，我们假设环境是长尾尾，也就是说，$ξ_n的分布$ f $：= \ log（（（1-a_n）/a_n）/a_n）$是长尾的。我们证明，尽管后代分布是轻尾的，但环境本身可以产生第n代人口规模的分布极为沉重的尾巴，随着n的增加而变得更重。更准确地说，我们证明，对于任何n，$ n $ th $ n $ th的人口尺寸$ z_n $的分销尾巴$ \ mathbb {p}（z_n> m）$在$ n \ overline {f}（\ log log m）上等同于$ m $ m $ m $ m $ grows。通过这种方式，我们概括了Bhattacharya和Palmowski（2019），他们在常规变化的环境$ f $的情况下证明了这一结果$ n = 1 $，参数$α> 1 $。此外，对于具有亚指数分布式$ξ_n$的亚临界分支过程，我们为分销尾巴提供渐近型$ \ mathbb {p}（z_n> m）$，对于所有$ n $，也适用于所有$ n $，也适用于常见的尾巴分配。然后，我们建立了“单个非典型环境的原理”，该原理说，颗粒数量较大的主要原因是单个非常小的环境参数$ a_k $的存在。
Motivated by a seminal paper of Kesten et al. (1975) we consider a branching process with a geometric offspring distribution with i.i.d. random environmental parameters $A_n$, $n\ge 1$ and size -1 immigration in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution $F$ of $ξ_n := \log ((1-A_n)/A_n)$ is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n-th generation which becomes even heavier with increase of n. More precisely, we prove that, for any n, the distribution tail $\mathbb{P}(Z_n > m)$ of the $n$-th population size $Z_n$ is asymptotically equivalent to $n\overline{F}(\log m)$ as $m$ grows. In this way we generalize Bhattacharya and Palmowski (2019) who proved this result in the case $n=1$ for regularly varying environment $F$ with parameter $α>1$. Further, for a subcritical branching process with subexponentially distributed $ξ_n$, we provide the asymptotics for the distribution tail $\mathbb{P}(Z_n>m)$ which are valid uniformly for all $n$, and also for the stationary tail distribution. Then we establish the "principle of a single atypical environment" which says that the main cause for the number of particles to be large is a presence of a single very small environmental parameter $A_k$.
