Almost sure local limit theorem for the Dickman distribution

We study the asymptotic behavior, and more precisely the second order properties, of the probabilistic model introduced in Hwang and Tsai (Comb Probab Comput 11(4):353–371, 2002) for describing the Dickman distribution. This model appears as an extremal example in the theory of the local and almost sure local limit theorem. We establish a delicate correlation inequality for this system. We apply it to obtain a fine almost sure local limit theorem. In doing so, we also give a corrected proof of the corresponding local limit theorem stated in Hwang and Tsai (Comb Probab Comput 11(4):353–371, 2002).