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arXiv · 2106.11279

Weighted Davis inequalities for martingale square functions

Dennis Wollgast·Pavel Zorin-Kranich
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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 \[ \mathbb{E} \Bigl( |f_0| w_0 + \frac{1}{4} \sum_{n=1}^{N} \frac{|df_n|^2}{f^*_n} w_n \Bigr) \leq \mathbb{E} ( f^*_N w^*_N). \] 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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Dennis Wollgast, Pavel Zorin-Kranich. 2021-06-21. Weighted Davis inequalities for martingale square functions. https://arxiv.org/abs/2106.11279

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