Skorokhod's topologies on path space

TL;DR

This paper extends Skorokhod's J1 and M1 topologies to handle convergence of cadlag functions on different domains by viewing their graphs as ordered compact sets, simplifying the theory and providing new proofs.

Contribution

It introduces a unified framework for J1 and M1 topologies based on graph orderings, extending their applicability and simplifying existing theory.

Findings

Extended topologies to functions on arbitrary closed subsets of the real line.

Proved the space of such paths is Polish.

Derived compactness criteria for the extended topologies.

Abstract

Skorokhod's J1 and M1 topologies are standard tools in proving limit theorems for stochastic processes. Motivated by applications, we extend these topologies so that they are capable of describing the convergence of a sequence of functions that are not all defined on the same domain. Traditionally, the J1 and M1 topologies are defined using time changes. Instead, we base our definitions on the point of view that the graph of a cadlag function can naturally be viewed as a compact set that is equipped with a total order. The distance between two graphs is then measured by matching points on one graph with points on the other graph in a way that respects the total order. We treat the J1 and M1 topologies in a unified framework and simplify the existing theory. We introduce a space of paths, elements of which are cadlag functions defined on an arbitrary closed subset of the real line. We show that this space is Polish and derive compactness criteria. Specialised to functions that are all defined on the same domain, this yields new proofs of known results.