Two Inequalities for the First Moments of a Martingale, its Square Function and its Maximal Function
Volume 53 / 2005
Bulletin Polish Acad. Sci. Math. 53 (2005), 441-449
MSC: Primary 60G42; Secondary 60G46.
DOI: 10.4064/ba53-4-9
Abstract
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.
