I'm looking at a Martingale (actually a Martingale difference sequence), $$ M_n = \sum \delta M_n, $$ and I'd like to prove something about convergence. If Martingale is Banach space valued (infinite dimensional), are there immediate analogs of the convergence theorems?

I'm looking at a Martingale (actually a Martingale difference sequence), $$ M_n = \sum \delta M_n, $$ and I'd like to prove something about convergence. If Martingale is Hilbert space valued (infinite dimensional), are there immediate analogs of the convergence theorems? I'm particularly interested in generalizing quadratic variation results to the Hilbert space case.