On the integrability of \(\sup | S_ n| /n^{1/r}\) for \(1<r<2\) (Q1091029)

Let \(X_ 1,X_ 2,..\). be i.i.d. random variables with \(EX_ 1=0\) and put \(S_ n=X_ 1+...+X_ n\). It is proved that for \(1<r<2\) the following three conditions are equivalent: \[ E(\sup_{n}n^{-1/r}| S_ n|)<\infty,\quad E(\sup_{n}n^{-1/r}| X_ n|)<\infty \quad and\quad E| X_ 1|^ r<\infty. \] It is stated that a straightforward modification of the proofs gives the equivalence, for \(1\leq r<2\), of the conditions \[ E(\sup_{n}n^{-1}| S_ n|^ r)<\infty,\quad E(\sup_{n}n^{-1}| X_ 1|^ r)<\infty \quad and\quad E\{| X_ 1|^ r(\log | X_ 1|)_+\}<\infty. \] Proofs are based on lemmas giving the equivalence of the condition for \(S_ n\) and the analogous condition for \(S_{n(k)}\) where \(n(k)<n(k+1)\) and \(\liminf_{k\to \infty}n(k-1)/n(k)>0\). \textit{A. Gut} [Ann. Probab. 7, 1059-1065 (1979; Zbl 0428.60036)] gives related results [see also \textit{A. Gut}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 46, 205-220 (1979; Zbl 0373.60059)].