A Non-Skorohod Topology on the Skorohod Space
Adam Jakubowski, Nicholas Copernicus University
Abstract
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 Skorohod'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 Prohorov's theorems
are valid, the a.s. Skorohod representation for subsequences exists and
finite dimensional convergence outside a countable set holds.
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