Finiteness of moments of randomly stopped sums of i.i.d. random variables (Q1121595)

Let \(\{X_ n\), \(n\geq 1\}\) be independent, identically distributed random variables such that \(E(X)=0\), and let \(Z_ n=\sum^{n}_{i=1}X_ i\). If N is a stopping time for \(\{X_ n\}\), \(Z_ N\) is called a randomly stopped sum. The main result of this paper states that for \(r\in (0,\infty)\) and \(a\in [1,2]\), \(E(N^{r/a})<\infty\) and \(E| X|^{\max (r,a)}<\infty\) together imply that \(E| Z_ N|^ r<\infty.\) This result generalizes some previous work by \textit{A. Gut} and \textit{S. Janson} [Ann. Probab. 14, 1296-1317 (1986; Zbl 0607.60055)].