Inventory model of type $(s,S)$ under heavy tailed demand with infinite variance

In this study, a stochastic process $X(t)$, which describes an inventory model of type $(s,S)$ is considered in the presence of heavy tailed demands with infinite variance. The aim of this study is observing the impact of regularly varying demand distributions with infinite variance on the stochastic process $X(t)$. The main motivation of this work is, the publication by Geluk [Proceedings of the American Mathematical Society125 (1997) 3407–3413] where he provided a special asymptotic expansion for renewal function generated by regularly varying random variables. Two term asymptotic expansion for the ergodic distribution function of the process $X(t)$ is obtained based on the main results proposed by Geluk [Proceedings of the American Mathematical Society125 (1997) 3407–3413]. Finally, weak convergence theorem for the ergodic distribution of this process is proved by using Karamata theory.