A Note on the Burkholder–Rosenthal Inequality
Volume 60 / 2012
Bulletin Polish Acad. Sci. Math. 60 (2012), 177-185
MSC: Primary 60G42; Secondary 60G46.
DOI: 10.4064/ba60-2-7
Abstract
Let $df$ be a Hilbert-space-valued martingale difference sequence. The paper is devoted to a new, elementary proof of the estimate $$ \left\|\sum_{k=0}^\infty df_k\right\|_p\leq C_p\left\{\left\|\left(\sum_{k=0}^\infty \mathbb E(|df_k|^2\,|\,\mathcal F_{k-1})\right)^{1/2}\right\| _p+\left\|\left(\sum_{k=0}^\infty |df_k|^p\right)^{1/p}\right\|_p\right\},$$ with $C_p=O(p/\!\ln p)$ as $p\to \infty$.
