Investigation of remainder terms in the ergodic distribution and moments of a renewal-reward process with heavy-tailed demand

This study investigated the detailed asymptotic behavior of the remainder terms in the ergodic distribution and its moments for a semi-Markovian renewal-reward process modeling an $(s,S)$-type inventory system. We focused on systems in which the demand random variables were heavy-tailed, specifically regularly varying with index $-\alpha$, where $1<\alpha <2$. While the first two terms in the asymptotic expansion of such models are available in the literature, earlier works have not provided sharp quantitative descriptions of the remainder. Our aim was to derive rigorous expressions that capture the exact decay of the remainder in both the ergodic distribution function and in the corresponding moments. Building on Doney's refinement of the renewal theorem ^[1], which distinguishes three main settings: the non-critical case $\alpha \ne 3/2$, the critical case $\alpha =3/2$ with square-integrable equilibrium distribution, and the case where such integrability fails, we established new asymptotic expansions that explicitly capture the decay structure of the remainder. Using this framework, we analyzed the remainder for both the ergodic distribution and its moments for each scenario.