Summen unabhängiger Zufallsvariablen, die durch die Maximalterme dominiert werden. (Sums of independent random variables which are dominated by maximal terms)
The author shows for i.i.d. random variables \(X_ i\) with a symmetric distribution function F that \[ (*)\;\limsup | X_ n| /(\max_{1\leq i\leq n-1}| S_ i|)=\infty \] if and only if \(EX^ 2_ 1=\infty\). In the other extreme case of positive random variables, an old result of Kesten states the equivalence of (*) with the non- existence of the first moment.
scientific article; zbMATH DE number 4209206