AI 中文总结
针对有限方差正则变化间隔分布的更新过程，建立二阶渐近展开式，揭示双尺度结构，分析 $\alpha=2$ 时的结构转变，为 $2\leq\alpha<3$ 提供统一描述并指出扩展方向。
AI 中文摘要
针对有限方差 regime 内具有正则变化间隔分布的更新过程，我们建立了精细的渐近理论。假设生存函数满足 $1-F(t)=t^{-\alpha}L(t)$，其中 $\alpha\in(2,3)$，$L$ 为慢变函数，我们对更新卷积和更新函数均建立了二阶渐近展开式。结果揭示了一种双尺度结构：平衡尾 $Q_F(t)$ 控制局部更新修正，而积分尾 $\rho(t)=\int_t^\infty Q_F(u)\\,du$ 决定了偏离平衡的全局偏差。我们特别关注临界阈值 $\alpha=2$，此处渐近行为会发生结构转变。在该边界情形下，幂律层级崩溃，主导修正由积分慢变尾决定。不过，局部更新修正机制依然存在，修正项仍渐近正比于平衡尾。分析基于拉普拉斯变换技术与陶伯型传递原理的系统应用，这些方法可将拉普拉斯域的奇异行为与精确的时域渐近相关联。所得结果为 $2\leq\alpha<3$ 整个范围内的二阶更新渐近提供了统一描述，并指出了向更新方程及年龄相关分支过程扩展的可能方向。
英文摘要
We develop a refined asymptotic theory for renewal processes with regularly varying inter-arrival distributions in the finite-variance regime. Assuming that $1-F(t)=t^{-α}L(t)$ with $α\in(2,3)$ 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 $ρ(t)=\int_t^\infty Q_F(u)\,du$ determines the global deviation from equilibrium. A particular emphasis is placed on the critical threshold $α=2$, 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 $2\leqα<3$, and point to possible extensions to renewal equations and to age-dependent branching processes.
