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A Detailed and Direct Proof of Skorohod-Wichura’s Theorem

Gane Samb Lo1,2,3*, Aladji Babacar Niang1 and Lois Chinwendu Okereke4

1LERSTAD, Gaston Berger University, Saint-Louis, Senegal.
2LSTA, Pierre and Marie Curie University, Paris VI, France.
3AUST – African University of Science and Technology, Abuja, Nigeria.
4Department of Pure and applied Mathematics, African University of Science and Technology (Abuja), Abuja, Nigeria.

Abstract

The representation Skorohod theorem of weak convergence of random variables on a metric space goes back to Skorohod (1956) in the case where the metric space is the class of real-valued functions defined on [0,1] which are right-continuous and have left-hand limits when endowed with the Skorohod metric. Among the extensions of that to metric spaces, the version by Wichura (1970) seems to be the most fundamental. But the proof of Wichura seems to be destined to a very restricted public. We propose a more detailed proof to make it more accessible at the graduate level. However we do far more by simplifying it since important steps in the original proof are dropped, which leads to a direct proof that we hope to be more understandable to a larger spectrum of readers. The current version is more appropriate for different kinds of generalizations.

Keywords: Skorohod-Wichura theorem; weak convergence of probability measure; product spaces; arbitrary products; complete spaces.

ISBN: 978-93-90149-72-8 (Print) | 978-93-90149-22-3 (eBook)

Chapter DOI: https://doi.org/10.9734/bpi/tpmcs/v1/5072D

Volume DOI: https://doi.org/10.9734/bpi/tpmcs/v1

Published: August 17, 2020

Complete book available here: http://bp.bookpi.org/index.php/bpi/catalog/book/237