A lattice renorming theorem and applications to vector-valued processes

A norm, $\vert\vert\;\vert\vert$, on a Banach space $E$ is said to be locally uniformly convex if $\left\Vert {{x_n}} \right\Vert \to \left\Vert x \right\Vert$ and $\left\Vert {{x_n} + x} \right\Vert \to 2\left\Vert x \right\Vert$ implies that ${x_n} \to x$ in norm. It is shown that a Banach lattice has an (order) equivalent locally uniformly convex norm if and only if the lattice is order continuous. This result is used to reduce convergence theorems for (lattice-valued) positive martingales and submartingales to the scalar case.