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

2018/09/06 by Li, Chris Junchi
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Probability (math.PR)

paper · doi:10.48550/arxiv.1809.02495

Abstract

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.

Related

• A Variant of Azuma's Inequality for Martingales with Subgaussian Tails
• On the Missing Factor in Some Concentration Inequalities for Martingales
• Time-uniform Chernoff bounds via nonnegative supermartingales
• New insights on concentration inequalities for self-normalized martingales
• Bernstein-type and Bennett-type inequalities for unbounded matrix martingales