On moment conditions for the supremum of normed sums (Q1094738)

The main result: Let \((X_ n)_{n\geq 1}\) be a sequence of random variables and \(c_ n\), \(n\geq 1\), constants with \(0<c_ n\to \infty\). If \(\sum_{n}E| X_ n|^{\alpha \beta}/c_ n^{\beta}<\infty\) for some \(\beta >1\) and \(0<\alpha \beta <2\), then \[ E\quad (\sup_{n}| \sum_{k\leq n}(X_ k-\alpha_ k)|^{\alpha}/c_ n)<\infty \] where \(\alpha_ k=0\) if \(0<\alpha \beta \leq 1\) and \(\alpha_ k=E(X_ k| X_ 1,...,X_{k-1})\) if \(1<\alpha \beta \leq 2\). This result is applied to the special case of i.i.d. \(X_ n's\).