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

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.