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]; 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.
Iosif Pinelis
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