Geodesic
Moment Inequality for the Martingale Square Function
Adam Osękowski1
1 Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland
Bulletin of the Polish Academy of Sciences. Mathematics, Tome 61 (2013) no. 2, pp. 169-180.

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Consider the sequence $(C_n)_{n\geq 1}$ of positive numbers defined by $C_1=1$ and $C_{n+1}=1+C_n^2/4$, $n=1,2,\ldots.$ Let $M$ be a real-valued martingale and let $S(M)$ denote its square function. We establish the bound $$ \mathbb E |M_n|\leq C_n\mathbb E S_n(M),\ \quad n=1,2 ,\ldots, $$ and show that for each $n$, the constant $C_n$ is the best possible.
DOI : 10.4064/ba61-2-11
Keywords: consider sequence geq positive numbers defined ldots real valued martingale denote its square function establish bound mathbb leq mathbb m quad ldots each constant best possible

Adam Osękowski 1

1 Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland 
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Adam Osękowski. Moment Inequality for the Martingale Square Function. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 61 (2013) no. 2, pp. 169-180. doi : 10.4064/ba61-2-11. http://geodesic.mathdoc.fr/articles/10.4064/ba61-2-11/

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