Let \(X_1,X_2,\dots\) be a sequence of i.i.d. random variables with mean \(0\) and partial sums \(\{S_n, n\geq 1\}\). The law of large numbers is concerned with the probabilities \(P(|S_n|>\varepsilon n)\) as \(n\to\infty\). In order to study convergence rates, the classical Hsu-Robbins-Spitzer-Erdős-Baum-Katz results establish necessary and sufficient conditions on the moments in order for sums like \(\sum^\infty_{n=1} P(|S_n|> \varepsilon n)\) to converge. The present paper is devoted to another aspect, namely the limiting behaviour of the sums, in terms of \(\varepsilon\), as \(\varepsilon\to 0\). More precisely, the authors prove limit theorems for \(\sum^\infty_{n=1} n^{r/p- 2}P(|S_n|> \varepsilon n^{1/p})\), properly normalized, as \(\varepsilon\searrow 0\).