Weighted Davis inequalities for martingale square functions

2021/06/21 by Wollgast, Dennis, Zorin-Kranich, Pavel
#60G42 (Primary) 60E15 #60G46 (Secondary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2106.11279

Abstract

For a Hilbert space valued martingale (f_n) and an adapted sequence of positive random variables (w_n), we show the weighted Davis type inequality 𝔼 ( |f₀| w₀ + (1)/(4) ∑_n=1^N (|df_n|²)/(f^*_n) w_n ) ≤ 𝔼 ( f^*_N w^*_N). This inequality is sharp and implies several results about the martingale square function. We also obtain a variant of this inequality for martingales with values in uniformly convex Banach spaces.

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