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

Abstract

In this article, we introduce an infinite-dimensional analogue of the $\alpha$-stable Lévy motion, defined as a Lévy process $Z=\{Z(t){\}}_{t\ge 0}$ with values in the space $\mathbb{D}$ of càdlàg functions on $[0,1]$, equipped with Skorokhod's $J_1$ topology. For each $t\ge 0$, $Z(t)$ is an $\alpha$-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 $\{S_n(t)={\sum }_{i=1}^{[nt]}{X}_i{\}}_{t\ge 0}$, suitably normalized and centered, associated to a sequence $(X_i{)}_{i\ge 1}$ of i.i.d. regularly varying elements in $\mathbb{D}$.

45 pages, 6 figures