Tightness of pairs of tight càdlàg processes (Q1073449)

A general theorem on necessary and sufficient conditions for the tightness of a sequence of cadlag processes is obtained. This theorem is used to derive necessary and sufficient conditions for the tightness in \({\mathcal D}([0,\infty), E_ 1\times E_ 2)\) of a sequence of pairs of cadlag processes which are individually tight in \({\mathcal D}([0,\infty),E_ i)\), \(i=1,2.\) Finally these results are applied to point processes and their compensators and the following result is proved: if the point processes and their compensators are individually tight, then their differences (i.e. the martingales given by the Doob-Meyer decomposition) form a tight sequence.