Markov Chain and Renewal Rates of Convergence via Nonparametric Structural Orderings ROBERT B. LUND University Of Georgia  We consider the problem of finding good geometric convergence rates for discrete renewal sequences and Markov chains.  The goal is to identify an explicit rate bound and first constant that can be computed via minimal information.  A general renewal convergence rate is first derived from the hazard rates of the renewal lifetimes.  This result is used to derive renewal convergence rates for lifetimes possessing the nonparametric new worse than used, new better than used, increasing  hazard rate, decreasing hazard rate, and stochastically monotone structures. The results are perhaps reminiscent of Barlow and Proschan.  Attention is then directed to Markov chain convergence issues.