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Extensions of Skorohod’s almost sure representation theorem

A well known result in probability is that convergence almost surely (a.s.) of a sequence of random elements implies weak convergence of their laws. The Ukrainian mathematician Anatoliy Volodymyrovych Skorohod proved the lemma known as Skorohod’s a.s. representation Theorem, a partial converse of this result. In this work we discuss the notion of continuous representations, which allows us to provide generalizations of Skorohod’s Theorem. In Chapter 2, we explore Blackwell and Dubins’s extension [3] and Fernique’s extension [10]. In Chapter 3 we present Cortissoz’s result [5], a variant of Skorokhod’s Theorem. It is shown that given a continuous path in M ( S ) it can be associated a continuous path with ﬁxed endpoints in the space of S -valued random elements on a nonatomic probability space, endowed with the topology of convergence in probability. In Chapter we modify Blackwell and Dubins representation for particular cases of S , such as certain subsets of R or R .