2021/02/09 by Louis H. Y. Chen, Chen, Louis H. Y., Arturo Jaramillo +3 · 1 citation
Mathematics · #11K65 #11N60 #60F05 #62E17 #Benford’s Law and Fraud Detection #FOS: Mathematics #Number Theory (math.NT) #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2102.05094
openalex publication_date 2021/02/09 · openalex created_date 2021/02/15 · openalex updated_date 2026/07/28
We present a new perspective of assessing the rates of convergence to the Gaussian and Poisson distributions in the Erdös-Kac theorem for additive arithmetic functions ψ of a random integer Jn 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 ψ 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 ψ(Jn) and the standard Gaussian distribution, and (ii) bounds on the Kolmogorov distance and total variation distance between the distribution of ψ(Jn) and a Poisson distribution under mild additional assumptions on ψ. Our results generalize the existing ones in the literature.