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Renewal function asymptotics refined à la Feller

Daley D.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 2

291--298

Feller’s volume 2 shows how to use the Key Renewal Theorem to prove that in the limit x → ∞, the renewal function U(x) of a renewal process with nonarithmetic generic lifetime X with finite mean E(X) = 1/λ and second moment differs from its linear asymptote λx by the quantity 1/2λ2E(X2). His first edition (1966) (but not the second in 1971) asserted that a similar approach would refine this asymptotic result when X has finite higher order moments. The paper shows how higher order moments may justify drawing conclusions from a recurrence relation that exploits a general renewal equation and further appeal to the Key Renewal Theorem.