On the maximal inequality
Let \(X_1\), \(X_2, \dots, X_n,\dots\) be a sequence of real i.i.d. random variables, \(S_n: =\sum^n_{k=1} X_k\), \(\lambda>0\) a real number, \(h>0\) a non-decreasing, convex real function, then \[ P\Bigl(\sup_{k\geq n}S_k\geq\varepsilon\Bigr)\leq Eh(\lambda S_n)/h (\lambda \varepsilon n) \] for every real \(\varepsilon>0\) and every integer \(n\geq 1\). This is proved using a reverse-martingale inequality and a moment estimate. Corollaries sharpen among others Hoeffding's and Banjevic's inequalities.
scientific article; zbMATH DE number 1031726
scientific article