  We present a new perspective of assessing the rates of convergence to the
Gaussian and Poisson distributions in the Erd\"os-Kac theorem for additive
arithmetic functions $\psi$ of a random integer $J_n$ uniformly distributed
over $\{1,...,n\}$. Our approach is probabilistic, working directly on spaces
of 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 the
distribution of the normalized $\psi(J_n)$ and the standard Gaussian
distribution, and (ii) bounds on the Kolmogorov distance and total variation
distance between the distribution of $\psi(J_n)$ and a Poisson distribution
under mild additional assumptions on $\psi$. Our results generalize the
existing ones in the literature.
