Which i.i.d. sums are recurrently dominated by their maximal terms? (Q1210335)

Let \((X_ n)_{n\geq 1}\) be i.i.d. random variables with common distribution function \(F\) and \(P(X_ 1\neq 0)>0\). Put \(S_ n=\sum^ n_{i=1} X_ i\), \(S^*_ 0=0\), \(S^*_ n=\sup_{1\leq i\leq n}| S_ i|\), \(n\geq 1\). The authors characterize \(\limsup_{n\to\infty} (X_ n/S^*_{n-1})=\infty\) a.s. in terms of the distribution function \(F\). It turns out that their results are generalizations of previous results by \textit{H. Kesten} [Ann. Math. Stat. 41, 1173-1205 (1970; Zbl 0233.60062)] and \textit{R. Wittmann} [Acta Math. Hung. 56, No. 3/4, 225-228 (1990; Zbl 0731.60025)].