A Note on the Men'shov–Rademacher Inequality
Volume 54 / 2006
Bulletin Polish Acad. Sci. Math. 54 (2006), 89-93
MSC: Primary 26D15; Secondary 60E15.
DOI: 10.4064/ba54-1-9
Abstract
We improve the constants in the Men'shov–Rademacher inequality by showing that for $n\ge 64$, $$ \textbf{E}\Big(\sup_{1\le k\le n}\Big|\sum^k_{i=1} X_i\Big|^2\Big)\le 0.11(6.20+\log_2 n)^2 $$ for all orthogonal random variables $X_1,\ldots ,X_n$ such that $\sum^n_{k=1}\textbf{E}|X_k|^2=1$.
