A remark on Dubins-Savage inequality (Q1185542)

Let \(\{X_ n,{\mathcal F_ n},\;n\geq 0\}\) be a martingale difference sequence such that \(E(X_ n\mid{\mathcal F}_{n-1})=0\) and \(V_ n=E(X^ 2_ n\mid{\mathcal F}_{n-1})\) is finite, where \(V_ 1\) is constant. Set \(S_ n=X_ 1+X_ 2+\dots+X_ n\) and let \(a>0\), \(b>0\). The paper proves the following inequalities and discusses some of their consequences: \[ P\left(S_ n\geq b+a\sum_{i=1}^ n V_ i\text{ for some } n\geq 1\right)\leq \min[(1+ab)^{-1},(1+V_ 1a^ 2/4)^{-1}] \] and \[ P\left(| S_ n|\geq b+a\sum_{i=1}^ n V_ i\text{ for some } n\geq 1\right)\leq 2 \min[(1+ab)^{-1},(1+V_ 1a^ 2/4)^{- 1}]. \]