A non-Skorokhod topology on the Skorokhod space. (English) Zbl 0890.60003

Jakubowski, Adam

Electron. J. Probab. 2, Paper 4, 21 p. (1997).

Summary: A new topology (called \(S\)) is defined on the space \(D\) of functions \(x: [0,1] \to R^1\) which are right-continuous and admit limits from the left at each \(t > 0\). Although \(S\) cannot be metricized, it is quite natural and shares many useful properties with the traditional Skorokhod’s topologies \(J_1\) and \(M_1\). In particular, on the space \(P(D)\) of laws of stochastic processes with trajectories in \(D\) the topology \(S\) induces a sequential topology for which both the direct and the converse Prokhorov’s theorems are valid, the a.s. Skorokhod representation for subsequences exists and finite-dimensional convergence outside a countable set holds.

MSC:

60B05	Probability measures on topological spaces
60F17	Functional limit theorems; invariance principles
60G17	Sample path properties
54D55	Sequential spaces

Keywords:

Skorokhod space; Skorokhod representation; convergence in distribution; sequential spaces; semimartingales