Jacod, Jean; Protter, Philip A remark on the weak convergence of processes in the Skorohod topology. (English) Zbl 0787.60007 J. Theor. Probab. 6, No. 3, 463-472 (1993). 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 \(\infty\). Reviewer: R.Norvaiša (Ottawa) Cited in 1 Document MSC: 60B10 Convergence of probability measures 60F17 Functional limit theorems; invariance principles 60G40 Stopping times; optimal stopping problems; gambling theory Keywords:Skorokhod topology; random time changes PDFBibTeX XMLCite \textit{J. Jacod} and \textit{P. Protter}, J. Theor. Probab. 6, No. 3, 463--472 (1993; Zbl 0787.60007) Full Text: DOI References: [1] Billingsley, P. (1968).Convergence of Probability Measures, Wiley, New York. · Zbl 0172.21201 [2] Dellacherie, C. (1972).Capacit?s et Processes Stochastiques, Springer-Verlag, Berlin/Heidelberg/New York. · Zbl 0246.60032 [3] Ethier, S., and Kurtz, T. G. (1986).Markov Processes: Characterization and Convergence, Wiley, New York. · Zbl 0592.60049 [4] Himmelberg, C. J. (1975). Measurable relations,Fund. Math. 87, 53-72. · Zbl 0296.28003 [5] Jacod, J., and Shiryaev, A. N. (1987)Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin/Heidelberg/New York. · Zbl 0635.60021 [6] Pollard, D. (1984).Convergence of Stochastic Processes, Springer-Verlag, Berlin/Heidelberg/New York. · Zbl 0544.60045 [7] Wagner, D. H. (1977). Survey of measurable selection theorems.SIAM J. Control and Optimization 15, 859-903. · Zbl 0407.28006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.