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

Electronic Communications in Probability

Adam Jakubowski¹ 

关键词: functional convergence of stochastic processes;    $S$ topology;    $J_1$ topology;    Skorokhod space;    sequential spaces;   

DOI  :  10.1214/17-ECP105

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

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.

【 授权许可】

CC BY