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A Note on the Burkholder–Rosenthal Inequality

Volume 60 / 2012

Adam Osękowski Bulletin Polish Acad. Sci. Math. 60 (2012), 177-185 MSC: Primary 60G42; Secondary 60G46. DOI: 10.4064/ba60-2-7


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 $p\to \infty$.


  • Adam OsękowskiFaculty of Mathematics, Informatics and Mechanics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland

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