Are any concentration inequality available for $d$-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension $d$ in the inequality. 


Thanks. 



concentration inequality for $d$-dimensional martingale
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](http://epubs.siam.org/doi/abs/10.1137/S0040585X97T987417)]**; 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; cf. e.g. the way Bernstein's exponential bound is derived from bounds on moments. Alternatively (and sometimes more efficiently), instead of deriving bounds on the exponential moments, one can use, more directly, an inequality of the form
$$P(X\ge x)\le\inf\_{p>p\_0}E|X|^p/x^p
$$
for $x>0$ and $p\_0\ge0$.


