Bounds (not depending explicitly on the dimension) on the moments of the norm of martingales in arbitrary 2-smooth Banach spaces (which of course include all finite-dimensional Euclidean spaces) can be found in **[[1][1]]**; see also further references there. As usual, under appropriate conditions, such bounds on the moments imply the corresponding bounds on the tails of the distributions of such martingales. 


  [1]: http://epubs.siam.org/doi/abs/10.1137/S0040585X97T987417