A note on concentration inequality for vector-valued martingales with weak exponential-type tails
This provides theoretical tools for statistical applications involving heavy-tailed data, but it is incremental as it builds on existing work.
The paper tackles the problem of deriving concentration inequalities for vector-valued martingales with weak exponential-type tails, proving that tail bounds depend on the sum of squared Orlicz-ψα norms rather than the maximal norm, generalizing prior results.
We present novel martingale concentration inequalities for martingale differences with finite Orlicz-$ψ_α$ 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-$ψ_α$ norms instead of the maximal Orlicz-$ψ_α$ norm, generalizing the results of Lesigne & Volný (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.