Two Inequalities for the First Moments of a Martingale, its Square Function and its Maximal Function

Volume 53 / 2005

Adam Osękowski Bulletin Polish Acad. Sci. Math. 53 (2005), 441-449 MSC: Primary 60G42; Secondary 60G46. DOI: 10.4064/ba53-4-9


Given a Hilbert space valued martingale $(M_n)$, let $(M^*_n)$ and $(S_n(M))$ denote its maximal function and square function, respectively. We prove that $$\displaylines{ \mathbb{E}|M_n|\leq 2\mathbb{E}S_n(M), \quad\ n=0,1,2,\ldots,\cr \mathbb{E}M^*_n \leq \mathbb{E}|M_n|+2\mathbb{E}S_n(M), \quad\ n=0,1,2,\ldots. } $$ The first inequality is sharp, and it is strict in all nontrivial cases.


  • Adam OsękowskiInstitute of Mathematics
    Warsaw University
    Banacha 2
    02-097 Warszawa, Poland

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