Given an additive function f let \(f_ k\) (k\(\geq 1)\) be the associated ''truncated'' functions defined by \(f_ k(n)=\sum_{p^ m\| n, p\leq k}f(p^ m).\) The author first characterizes those additive functions f, for which the sequence \((f_ k)\) converges strongly to f in the sense that for every \(\epsilon >0\) \[ \lim_{k\to \infty} \limsup_{x\to \infty}(1/x) \#\{n\leq x:\quad | f_ k(n)-f(n)| \geq \epsilon \}=0 ; \] the criterion for this coincides with the Erdős-Wintner criterion for the existence of a limit distribution for f. He then proves similar results involving renormalizing constants. Moreover, he proves an analogue of the law of iterated logarithm for the sequence \((f_ k)\), subject to suitable conditions on the function f.