Two new proofs of the Erd_s_Kac Theorem, with bound on the rate of convergence, by Stein's method for distributional approximations

摘要

In this paper, we apply Stein's method for distributional approximations to prove a quant_itative form of the Erd_s-Kac Theorem. We obtain our best bound on the rate of conver_gence, on the order of log log log n (log log n)~(-1/2), by making an intermediate Poisson ap_proximation; we believe that this approach is simpler and more probabilistic than others, and we also obtain an explicit numerical value for the constant implicit in the bound. Dif_ferent ways of applying Stein's method to prove the Erd_s-Kac Theorem are discussed, including a Normal approximation argument via exchangeable pairs, where the suitability of a Poisson approximation naturally suggests itself.