Moment Inequality for the Martingale Square Function
Volume 61 / 2013
Bulletin Polish Acad. Sci. Math. 61 (2013), 169-180
MSC: Primary 60G42; Secondary 60E15.
DOI: 10.4064/ba61-2-11
Abstract
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.
