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Stable Lévy Motion with Values in the Skorokhod Space: Construction and Approximation

  • Raluca M. BalanEmail author
  • Becem Saidani
Article
  • 21 Downloads

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(ts) 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 (Ann Probab 29:467–483, 2001) and Hult and Lindskog (Stoch Proc Appl 115:249–274, 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 with a sequence \((X_i)_{i\ge 1}\) of i.i.d. regularly varying elements in \({\mathbb {D}}\).

Keywords

Functional limit theorems Skorokhod space Lévy processes Regular variation 

Mathematics Subject Classification

Primary 60F17 Secondary 60G51 60G52 

Notes

Acknowledgements

We would like to thank François Roueff, Gennady Samorodnitsky and Philippe Soulier for useful discussions, and for drawing our attention to reference [8] regarding \(\alpha \)-stable Lévy processes on cones (see Remark 1.2). We are also grateful to Thomas Mikosch for the proof of Lemma C.1, to Xiao Liang for his help with the simulations and to Adam Jakubowski for reading the manuscript.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

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