A remark on the weak convergence of processes in the Skorohod topology (Q685730)

It is proved that if stochastic processes \(X_ n\) converge to a stochastic process \(X\) in the Skorokhod topology, then there exist random time changes \(\Lambda_ n\) that are also stochastic processes and such that \(X_ n \circ \Lambda_ n\) converges to \(X\) in the appropriate sense. Moreover, if the processes \(X_ n\), \(X\) are defined on the same space, and letting \({\mathcal F}_ n(t)=\sigma(X_ n(s),\;X(s):s \leq t)\), \(t \geq 0\), denote the natural filtration, one can choose the \(\Lambda_ n\) such that they are adapted to \({\mathcal F}_ n (t+\gamma_ n)\), \(t \geq 0\), where \(\gamma_ n\) is a sequence of constants decreasing to 0 as \(n\) tends to \(\infty\).