Random variables without first moment overtake their partial sums (Q2722490)

The authors prove the following result: Let \((X_1,X_2,\dots)\) be an i.i.d. sequence of strictly positive random variables on some probability space \((\Omega,{\mathcal F},\mathbb{P})\) which fulfill the condition: There exist \(C_1> 0\) and \(C_2> 0\) such that NEWLINE\[NEWLINE{C_1\over 1+t}\leq \mathbb{P}(X_1> t)\leq {C_2\over 1+t},\quad\text{for every }t> 0.NEWLINE\]NEWLINE Then NEWLINE\[NEWLINE\sum^\infty_{n=1} {1\over X_1+\cdots+ X_n}= \infty\text{ a.s.}\quad\text{and}\quad \sum^\infty_{n=1} 1_{\{X_{n+1}> X_1+\cdots+ X_n\}}= \infty \text{ a.s}.NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0958.00034].