Convergence of weighted sums and laws of large numbers in \(D([0,1]; E)\)
Let \(D\) be the space of functions which are right-continuous and have left limits. It is shown that convergence in the Skorokhod \(J_1\)- topology of a sequence \((x_n)\) in \(D\) does not imply convergence of a sequence \((\overline x_n)\) of averages. However, convergence of a sequence \((x_n)\) is shown to imply convergence of the sequence \((\overline x_n)\) of averages in the \(M_1\) and \(M_2\) topologies. Finally, a sequence of independent \(D\)-valued random variables is constructed to show that the strong laws of large numbers with respect to \(J_1\) and uniform topologies are nonequivalent.
scientific article; zbMATH DE number 769839