Stable L\'evy motion with values in the Skorokhod space: construction and approximation

Abstract

In this article, we introduce an infinite-dimensional analogue of the $\alpha$-stable L\'evy motion, defined as a L\'evy process $Z=\{Z(t)\}_{t \geq 0}$ with values in the space $\mathbb{D}$ of c\`adl\`ag functions on $[0,1]$, equipped with Skorokhod's $J_1$ topology. For each $t \geq 0$, $Z(t)$ 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 c\`adl\`ag 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 $\mathbb{D}$.