On the Skorokhod representation theorem

Abstract:

In this paper we present a variant of the well-known Skorokhod Representation Theorem. First we prove, given $S$ a Polish Space, that to a given continuous path $\alpha$ in the space of probability measures on $S$, we can associate a continuous path in the space of $S$-valued random variables on a nonatomic probability space (endowed with the topology of the convergence in probability). We call this associated path a lifting of $\alpha$. An interesting feature of our result is that we can fix the endpoints of the lifting of $\alpha$, as long as their distributions correspond to the respective endpoints of $\alpha$. Finally, we also discuss and prove an $n$-dimensional generalization of this result.