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

Abstract

We present novel martingale concentration inequalities for martingale differences with finite Orlicz-_ 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-_ norms instead of the maximal Orlicz-_ norm, generalizing the results of Lesigne & Voln\'y (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.

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