Let X be a random variable and let \(\mu (x)=\int^{x}_{0}P[| X| >t]dt\) provided E \(| X| =\infty\) and \({\tilde \mu}(x)= \int^{\infty}_{x}P[| X| >t]dt\) when E \(| X| <\infty\). Assume that \(\mu\) and \({\tilde \mu}\) are slowly varying at infinity and denote by \(c_ x\) and \(\tilde c_ x\) the inverse functions of \(\mu\) and \({\tilde \mu}\), resp. Next, put \(S_ n=\sum^{n}_{k=1}k^{\alpha}X_ k\), \(b_ n=n^{\alpha}c_ n\) and \(\tilde b_ n=n^{\alpha}\tilde c_ n\). The following two main results are proved:
1. Let \(\{X,X_ n,n\leq 1\}\) be i.i.d. r.v.’s with E \(| X| =\infty\) and \(E=(X^-/\mu (X^-))<\infty\). If \(\mu\) (x)\(\sim \mu (x \log \log x)\), then for each \(\alpha >-1:\) \[ \liminf_{n\to \infty}S_ n/b_ n=(\alpha +1)^{-1}\text{ and } \limsup_{n\to \infty}S_ n/b_ n=\infty \quad a.s. \] 2. Let \(\{X,X_ n,n\geq 1\}\) be integrable, unbounded i.i.d. mean zero r.v.’s with \(E(X^-/{\tilde \mu}(X^- ))<\infty\). If \({\tilde \mu}\)(x)\(\sim {\tilde \mu}(x \log \log x)\) and \({\tilde \mu}(\tilde b_ n)=0({\tilde \mu}(\tilde c_ n))\) for \(\alpha\in (-1,-1/2)\), then \[ \liminf_{n\to \infty}S_ n/\tilde b_ n=-(\alpha +1)^{-1}\text{ and } \sup_{n\to \infty}S_ n/\tilde b_ n=\infty \quad a.s. \] whenever \(\alpha >-1\). Moreover, it is shown that if \(\{X,X_ n,n\geq 1\}\) are i.i.d. r.v.’s with E \(| X|^ p<\infty\) for all \(p<1\), then \(S_ n\) converges a.s. for each fixed \(\alpha <-1\), so that \(S_ n/b_ n\to 0\) with probability 1 whenver \(| b_ n| \to \infty\).