Geometric renewal convergence rates from hazard rates
Let \((u_n)\) be the renewal sequence for a positive integer random variable \(X\) with finite mean and unit span; so \(u_n\to u_\infty= (EX)^{-1}\). Known conditions for geometric convergence: \(|u_n- u_\infty|< \kappa r^{-n}\), \(n\geq 0\), for some rate \(r> 1\) and constant \(\kappa> 0\), are reviewed. Defining \(r_n= P(X> n-1)/P(X> n)\), with \(r_n= 1\) when \(P(X> n)= 0\), it is shown that \(r\) and \(\kappa^{-1}\) may be taken to be \(\inf_n r_n\); in particular \(|u_n- u_\infty|\leq (P(X> 1))^{n+ 1}\) for new better than used \(X\). This rate cannot be bettered in general, but for the IHR class (\(r_n\) strictly increasing) an improved rate is given, again in terms of \((r_n)\). Illustrate examples are provided, some of which discuss convergence rates for Markov chains.
scientific article; zbMATH DE number 1625616