Asymptotic enumeration by Khintchine–Meinardus probabilistic method: necessary and sufficient conditions for sub-exponential growth

In this paper, we prove the necessity of the main sufficient condition of Meinardus for sub-exponential rate of growth of the number of structures, having multiplicative generating functions of a general form and establish a new necessary and sufficient condition for normal local limit theorem for aforementioned structures. The latter result allows to encompass in our study structures with sequences of weights having gaps in their support.