Summary: Let \(S_ n=\sum^ n_{j=1}X_ j\), \(n \geq 1\), where \(\{X_ n,n \geq 1\}\) are i.i.d. random variables. A well-known result of M. Katz [Ann. Math. Stat. 39, 1348-1349 (1968; Zbl 0162.494)], which asserts that \(S_ n/n\) converging to 0 in probability but not almost certainly implies \(\lim\sup_{n\to\infty}S_ n/n=-\lim\inf_{n\to\infty}S_ n/n=\infty\) almost certainly, is generalized and its proof is considerably simplified. More specifically, it is shown that if \(\sup_{n \geq 1}P \{| S_ n |/n>M\}<1\) for some constant \(0<M<\infty\) and \(\lim_{n \to \infty}S_ n/n=c\) almost certainly holds for no constant \(-\infty<c<\infty\), then the same conclusion as in Katz’s theorem obtains. While Katz’s proof rests on a deep combinatorial result of F. Spitzer [Trans. Am. Math. Soc. 82, 323-339 (1956; Zbl 0071.130)], the current proof involves only standard techniques and avoids combinatorial foundation. It is also pointed out that the preceding two results do not hold for a general sequence of norming constants unless there are additional conditions. Finally, an example is provided showing that there can be no analogue of Katz’s theorem for weighted averages of i.i.d. random variables.