Becem Saidani | Searcharxiv

arXiv2019

Weak convergence and tightness of probability measures in an abstract Skorohod space

In this article, we introduce the space $D([0,1];D)$ of functions defined on $[0,1]$ with values in the Skorohod space $D$, which are right-continuous and have left limits with respect to the $J_1$ topology. This space is equipped with the Skorohod-type distance introduced in Whitt (1980). Following the classical approach of Billingsley (1968, 1999), we give several criteria for tightness of probability measures on this space, by characterizing the relatively compact subsets of this space. In particular, one of these criteria has been used in the recent article Balan and Saidani (2018) for proving the existence of a $D$-valued $α$-stable Lévy motion. Finally, we give a criterion for weak convergence of random elements in $D([0,1];D)$, and a criterion for the existence of a process with sample paths in $D([0,1];D)$ based on its finite-dimensional distributions.

math.PR↗

arXiv2018

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

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 $\mathbb{D}$.

math.PR↗