Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Skorohod topology for functions with values in a completely regular space

Svante Janson

TL;DR

The paper addresses the foundational issue of defining the Skorohod ($J_1$) topology for càdlàg functions valued in a completely regular space $E$ and fixes a gap in Jakubowski's 1986 proof that this topology depends only on the topology of $E$ (not the chosen pseudometrics). It provides a complete argument for the independence result, introduces the split interval $\widehat{I}$ to relate cadlag functions to continuous extensions, and derives a functoriality result showing that continuous maps $\psi:E\to F$ induce continuous maps between the corresponding Skorohod spaces. The contributions establish that $D([0,1],E)$ carries a well-defined topology independent of metric choices, with extensions to $D([0,\infty),E)$ and explicit corollaries clarifying continuity under compositions. Overall, the work strengthens the theoretical foundations of convergence in distribution for stochastic processes with values in general topological spaces, beyond metric targets.

Abstract

We correct a gap in the proof of a basic theorem by Jakubowski (1986) on the Skorohod topology on the space of functions on [0,1] with values in a completely regular topological space.

On the Skorohod topology for functions with values in a completely regular space

TL;DR

The paper addresses the foundational issue of defining the Skorohod () topology for càdlàg functions valued in a completely regular space and fixes a gap in Jakubowski's 1986 proof that this topology depends only on the topology of (not the chosen pseudometrics). It provides a complete argument for the independence result, introduces the split interval to relate cadlag functions to continuous extensions, and derives a functoriality result showing that continuous maps induce continuous maps between the corresponding Skorohod spaces. The contributions establish that carries a well-defined topology independent of metric choices, with extensions to and explicit corollaries clarifying continuity under compositions. Overall, the work strengthens the theoretical foundations of convergence in distribution for stochastic processes with values in general topological spaces, beyond metric targets.

Abstract

We correct a gap in the proof of a basic theorem by Jakubowski (1986) on the Skorohod topology on the space of functions on [0,1] with values in a completely regular topological space.

Paper Structure

This paper contains 3 sections, 5 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Skorohod topology on $D([0,1],E)$

Key Result

Lemma 2.3

If $E$ is a regular (T${}_3$) topological space, and $f:[0,1]\to E$ is a function, then the following are equivalent: If this holds, then the continuous extension to $\widehat{I}$ is unique, and is the natural extension given by ft-.

Theorems & Definitions (17)

  • Remark 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Corollary 2.5
  • proof
  • Example 2.6
  • ...and 7 more