SECOND-ORDER RENEWAL ASYMPTOTICS FOR HEAVY-TAILED LIFETIME DISTRIBUTIONS: A LAPLACE-TAUBERIAN APPROACH

For a renewal process whose inter-arrival distribution has a regularly varying tail of index with , this paper summarizes a unified Laplace-Stieltjes framework for second-order renewal asymptotics. The centered renewal function and the secondary residual are treated as successive terms of one geometric transform series, and Karamata theory is combined with tauberian-abelian inversion to obtain sharp asymptotics across the infinite-variance interior regime and the critical boundary , where the behavior bifurcates according to whether the second moment is finite. A worked Pareto example illustrates the results.

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