New characterizations of the S topology on the Skorokhod space

TL;DR

This paper provides new characterizations of the $S$ topology on the Skorokhod space, highlighting its compactness properties, local convexity, and its relation to other topologies, along with extensions to multidimensional and infinite time domains.

Contribution

It offers a comprehensive analysis of the $S$ topology, including its compact description, local convexity, and comparisons with other topologies, plus extensions to broader function spaces.

Findings

Sequences converge in $S$ topology with a compact characterization.

$S$ topology is finer than any linear topology coarser than $J_1$.

Extensions of $S$ topology to multidimensional and infinite domains are defined.

Abstract

The $S$ topology on the Skorokhod space was introduced by the author in 1997 and since then it proved to be a useful tool in several areas of the theory of stochastic processes. The paper brings complementary information on the $S$ topology. It is shown that the convergence of sequences in the $S$ topology admits a compact description, exhibiting the locally convex character of the $S$ topology. It is also shown that $S$ is, up to some technicalities, finer than any linear topology which is coarser than Skorokhod's $J_1$ topology. The paper contains also definitions of extensions of the $S$ topology to the Skorokhod space of functions defined on $[0,+\infty)$ and with multidimensional values.