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We present a new perspective of assessing the rates of convergence to theGaussian and Poisson distributions in the Erd\"os-Kac theorem for additivearithmetic functions $\psi$ of a random integer $J_n$ uniformly distributedover $\{1,...,n\}$. Our approach is probabilistic, working directly on spacesof random variables without any use of Fourier analytic methods, and our $\psi$is more general than those considered in the literature. Our main results are(i) bounds on the Kolmogorov distance and Wasserstein distance between thedistribution of the normalized $\psi(J_n)$ and the standard Gaussiandistribution, and (ii) bounds on the Kolmogorov distance and total variationdistance between the distribution of $\psi(J_n)$ and a Poisson distributionunder mild additional assumptions on $\psi$. Our results generalize theexisting ones in the literature.

A probabilistic approach to the Erd\"os-Kac theorem for additive functions