A note on concentration inequality for vector-valued martingales with weak exponential-type tails

We present novel martingale concentration inequalities for martingale differences with finite Orlicz-$psi_alpha$ norms. Such martingale differences with weak exponential-type tails scatters in many statistical applications and can be heavier than sub-exponential distributions. In the case of one dimension, we prove in general that for a sequence of scalar-valued supermartingale difference, the tail bound depends solely on the sum of squared Orlicz-$psi_alpha$ norms instead of the maximal Orlicz-$psi_alpha$ norm, generalizing the results of Lesigne & Volny (2001) and Fan et al. (2012). In the multidimensional case, using a dimension reduction lemma proposed by Kallenberg & Sztencel (1991) we show that essentially the same concentration tail bound holds for vector-valued martingale difference sequences.