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)