论文标题
限制在1DLévy随机介质上的Lévy航班定理
Limit theorems for Lévy flights on a 1D Lévy random medium
论文作者
论文摘要
我们研究了一个点的随机步行，该过程是由点$（ω_k，\，k \ in \ mathbb {z}）$在实际行中给出的点过程。距离$ω_{k+1} - ω_k$是i.i.d. $β$稳定定律的吸引域中的随机变量，$β\ in（0,1）\ cup（1,2）$。随机步行有I.I.D.跳高使得$ω_k$和$ω__\ ell $之间的过渡概率取决于$ \ ell-k $，并且通过分布$ \ mathbb {z} $的分布在吸引$α$稳定法的域中有价值的随机变量，该变量具有$α$ - 稳定性法，$α\ in（0,1,1）\ cup（1,1,2）$（1,1,2）。由于定义变量（对于随机步行和积分过程都是重尾），因此我们谈到了Lévy随机介质上的Lévy飞行。对于参数$α$和$β$的所有组合，我们证明了适当重新恢复过程的退火功能极限定理，相对于每种情况下的最佳Skorokhod拓扑。当限制过程不是càdlàg时，我们证明了有限维分布的融合。当极限过程确定性时，我们还证明了波动的极限定理，仅相对于最佳的Skorokhod拓扑。
We study a random walk on a point process given by an ordered array of points $(ω_k, \, k \in \mathbb{Z})$ on the real line. The distances $ω_{k+1} - ω_k$ are i.i.d. random variables in the domain of attraction of a $β$-stable law, with $β\in (0,1) \cup (1,2)$. The random walk has i.i.d. jumps such that the transition probabilities between $ω_k$ and $ω_\ell$ depend on $\ell-k$ and are given by the distribution of a $\mathbb{Z}$-valued random variable in the domain of attraction of an $α$-stable law, with $α\in (0,1) \cup (1,2)$. Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters $α$ and $β$, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.
