Product Topologies on the Space of Distribution Functions

We study the connection between weak convergence in the space $\Delta_r$ of multiple distribution functions and convergence in the product topology induced on the product $\Delta\times\dots\times\Delta$ by the metrics on these spaces. We show that, for r>1 , weak convergence in $\Delta_r$ is slightly more general than convergence in either product topology on $\Delta\times\dots\times\Delta$. We also give several sufficient conditions under which these two modes of convergence are equivalent