Convergence of weighted sums of random functions in \(D[0,1]\) (Q1175672)

Let \(D=D([0,1];E)\) be the space of càdlàg functions on \([0,1]\) with values in a separable Banach space \(E\). A sequence \((X_ n)\) of random functions in \(D\) is said to satisfy condition (mt) if, for every \(\epsilon>0\), there exists a partition \(\pi=\{I_ j\}\) of \([0,1]\) such that \[ \max_ j E\sup_{s,t\in I_ f}\parallel X_ n(s)-X_ n(t)\parallel<\epsilon,\qquad n\in \mathbb{N}. \] It is proved that the condition (mt) is sufficient for tightness in the Skorokhod \(J_ 1\)- topology of weighted sums \(S_ n=\sum_{i=1}^{k_ n}a_{ni}X_ i\) provided \[ \sum_{i=1}^{k_ n}| a_{ni}| \leq C,\qquad \forall n\in\mathbb{N},\qquad\hbox{ and} \qquad \max_{1\leq i\leq k_ n}| a_{ni}| \to 0,\hbox{ as }n\to \infty. \] Weak laws of large numbers for \((X_ n)\) are then obtained via Prokhorov's theorem. Finally, as a consequence sufficient conditions for \(J_ 1\)-tightness of \((S_ n)\) and, in particular, of the sequence \((\overline {X}_ n)\) of averages are given in terms of the Skorokhod \(M_ 1\)-topology.