Electronic Communications in Probability

New characterizations of the $S$ topology on the Skorokhod space

Adam Jakubowski

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The $S$ topology on the Skorokhod space was introduced by the author in 1997 and since then it has 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 closed form description, exhibiting the locally convex character of the $S$ topology. Morover, it is proved that the $S$ topology 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.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 2, 16 pp.

Received: 1 September 2016
Accepted: 28 December 2017
First available in Project Euclid: 9 January 2018

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Primary: 60F17: Functional limit theorems; invariance principles 60B11: Probability theory on linear topological spaces [See also 28C20] 54D55: Sequential spaces 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)

functional convergence of stochastic processes $S$ topology $J_1$ topology Skorokhod space sequential spaces

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Jakubowski, Adam. New characterizations of the $S$ topology on the Skorokhod space. Electron. Commun. Probab. 23 (2018), paper no. 2, 16 pp. doi:10.1214/17-ECP105. https://projecteuclid.org/euclid.ecp/1515467250

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