Second--order renewal asymptotics in the finite--variance regularly varying regime
We develop a refined asymptotic theory for renewal processes with regularly varying inter-arrival distributions in the finite-variance regime. Assuming that $1-F(t)=t^{-\alpha}L(t)$ with $\alpha\in(2,3)$ and $L$ slowly varying, we establish second-order asymptotic expansions for both the renewal convolution and the renewal function. The results reveal a two-scale structure: the equilibrium tail $Q_F(t)$ governs the local renewal correction, while the integrated tail $\rho(t)=\int_t^\infty Q_F(u)\,du$ determines the global deviation from equilibrium. A particular emphasis is placed on the critical threshold $\alpha=2$, where the asymptotic behaviour undergoes a structural transition. In this borderline case, the power-law hierarchy collapses and the dominant correction is governed by an integrated slowly varying tail. Nevertheless, the local renewal correction mechanism persists, and the correction term remains asymptotically proportional to the equilibrium tail. The analysis is based on a systematic use of Laplace-transform techniques and Tauberian transfer principles, which allow us to relate singular behaviour in the Laplace domain to precise time-domain asymptotics. The results provide a unified description of second-order renewal asymptotics across the entire range $2\leq\alpha<3$, and point to possible extensions to renewal equations and to age-dependent branching processes.