arXiv 1809.021032018-09-06math.PR

Skorokhod空间中稳定Lévy运动的构造与逼近

Stable Lévy motion with values in the Skorokhod space: construction and approximation

Raluca M. Balan, Becem Saidani

备注：: 45 pages, 6 figures

摘要：

在这篇文章中，我们介绍了一个无限维模拟的$α$稳定的Lévy运动，定义为一个Lévy过程$Z=\{Z(t)\}_{t \geq 0}$的值在空间$\mathbb{D}$的càdlàg功能$[0,1]$，配备Skorokhod的$J_1$拓扑。对于每个$t \geq 0$，$Z(t)$是一个$α$稳定过程，样本路径在$\mathbb{D}$中，用$\{Z(t,s)\}_{s\in [0,1]}$表示.直观地，$Z(t,s)$给出了过程$Z$在时间$t$和空间位置$s$处的值。该过程与de Haan和Lin（2001）以及Hult和Lindskog（2005）中引入的$\mathbb{D}$中随机元素的规则变化概念密切相关。给出了基于Poisson随机测度的$Z$的一种构造，并证明了$Z$有一个修改，其样本路径是$[0,\infty)$上的càdlàg函数，其值在$\mathbb{D}$中.最后，我们证明了一个功能性的极限定理，它确定了分布的这种修改的限制的部分和序列$\{S_n(t)=\sum_{i=1}^{[nt]}X_i\}_{t\geq 0}$，适当规范化和中心，关联到一个序列$(X_i)_{i\geq 1}$的i.i.d. $\mathbb{D}$中规则变化的元素。

英文摘要：

In this article, we introduce an infinite-dimensional analogue of the $α$-stable Lévy motion, defined as a Lévy process $Z=\{Z(t)\}_{t \geq 0}$ with values in the space $\mathbb{D}$ of càdlàg functions on $[0,1]$, equipped with Skorokhod's $J_1$ topology. For each $t \geq 0$, $Z(t)$ is an $α$-stable process with sample paths in $\mathbb{D}$, denoted by $\{Z(t,s)\}_{s\in [0,1]}$. Intuitively, $Z(t,s)$ gives the value of the process $Z$ at time $t$ and location $s$ in space. This process is closely related to the concept of regular variation for random elements in $\mathbb{D}$ introduced in de Haan and Lin (2001) and Hult and Lindskog (2005). We give a construction of $Z$ based on a Poisson random measure, and we show that $Z$ has a modification whose sample paths are càdlàg functions on $[0,\infty)$ with values in $\mathbb{D}$. Finally, we prove a functional limit theorem which identifies the distribution of this modification as the limit of the partial sum sequence $\{S_n(t)=\sum_{i=1}^{[nt]}X_i\}_{t\geq 0}$, suitably normalized and centered, associated to a sequence $(X_i)_{i\geq 1}$ of i.i.d. regularly varying elements in $\mathbb{D}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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