Renewal convergence rates for DHR and NWU lifetimes (Q2784123)

Consider a discrete renewal sequence \(\{u_n\}^\infty_{n=0}\) generated from independent and identically distributed lifetimes \(\{X_i\}^\infty_{i=1}\). It is assumed that the lifetimes are supported on a subset (but not a sublattice) of \(\{1,2,\dots\}\), that \(P[X_1= 1]< 1\), and that \(E[X_1]<\infty\). Then \(u_n\) is the probability that a recurrent event occurs at time \(n\); the convention \(u_0=1\) is made. The familiar renewal convergence theorem states that \(u_n\to E[X_1]^{-1}:= u_\infty\) as \(n\to\infty\). The goal of the paper is to study the geometric convergence rate of \(u_n\) to \(u_\infty\); in particular, the authors want to identify a geometric rate \(r>1\) and a constant \(\kappa<\infty\) satisfying \(|u_n- u_\infty|\leq \kappa r^{-n}\), \(n\geq 0\), and, whenever possible, to identify the largest \(r> 1\) possible. Good geometric convergence rates are derived when the distribution of \(X_1\) is DHR (decreasing hazard rate), IMRL (increasing mean residual life), and NWU (new worse than used). The authors also consider Markov chain convergence rates for chains with a DHR state.