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Geometric renewal convergence rates from hazard rates

Part of: Limit theorems Special processes

Published online by Cambridge University Press:  14 July 2016

Kenneth S. Berenhaut  and
Robert Lund
Kenneth S. Berenhaut*
Affiliation:
University of Georgia
Robert Lund*
Affiliation:
University of Georgia
*
∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602-1952, USA.
∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602-1952, USA.
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Abstract

This paper studies the geometric convergence rate of a discrete renewal sequence to its limit. A general convergence rate is first derived from the hazard rates of the renewal lifetimes. This result is used to extract a good convergence rate when the lifetimes are ordered in the sense of new better than used or increasing hazard rate. A bound for the best possible geometric convergence rate is derived for lifetimes having a finite support. Examples demonstrating the utility and sharpness of the results are presented. Several of the examples study convergence rates for Markov chains.

Keywords

geometric convergencerenewal sequenceMarkov chainspower serieshazard ratesincreasing hazard ratenew better than used

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 180 - 194
Copyright
Copyright © by the Applied Probability Trust 2001 

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References

Berenhaut, K. S., and Lund, R. B. (1999). Renewal convergence rates for DHR and NWU lifetimes. Submitted.Google Scholar
Berenhaut, K. S., and Lund, R. B. (2000). Tech. Rept 2000-13, Department of Statistics, University of Georgia.Google Scholar
Berbee, H. (1987). Convergence rates in the strong law for bounded mixing sequences. Prob. Theory Rel. Fields. 74, 255–270.Google Scholar
Brown, M. (1980). Bounds, inequalities, and monotonicity properties for some specialized renewal processes. Ann. Prob. 8, 227–240.Google Scholar
Bryson, M. C., and Siddiqui, M. M. (1969). Some criteria for aging. J. Amer. Statist. Assoc. 64, 1472–1483.Google Scholar
Davies, P. L., and Grubel, R. (1981). Spaces of summable sequences in renewal theory and the theory of Markov chains. Math. Nachr. 104, 119–128.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. John Wiley, New York.Google Scholar
Gelfond, A. O. (1964). An estimate for the remainder term in the limit theorem for recurrent events. Theory Prob. Appl. 9, 299–303.Google Scholar
Grubel, R. (1983). Functions of discrete probability measures: rates of convergence in the renewal theorem. Z. Wahrscheinlichkeitsth. 64, 341–357.Google Scholar
Hansen, B. G., and Frenk, J. B. G. (1991). Some monotonicity properties of the delayed renewal function. J. Appl. Prob. 28, 811–821.Google Scholar
Heathcote, C. R. (1967). Complete exponential convergence and related topics. J. Appl. Prob. 4, 1–40.Google Scholar
Kendall, D. G. (1959). Unitary dilations of Markov transition operators and the corresponding integral representations for transition probability matrices. In Probability and Statistics, ed. Grenander, U. John Wiley, New York, pp. 139–161.Google Scholar
Kijima, M. (1997). Markov Processes for Stochastic Modeling. Chapman and Hall, London.Google Scholar
Konstantopoulos, T., and Last, G. (1999). On the use of Lyapunov function methods in renewal theory. Stoch. Proc. Appl. 79, 165–178.Google Scholar
Liggett, T. (1989). Total positivity and renewal theory. In Probability, Statistics, and Mathematics. eds. Anderson, T. W., Athreya, K. B. and Iglehart, D. L. Academic Press, Boston, pp. 141–162.Google Scholar
Lindvall, T. (1979). On coupling of discrete renewal processes. Z. Wahrscheinlichkeitsth. 48, 57–70.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
Lund, R. B., and Tweedie, R. L. (1996). Geometric convergence rates for stochastically ordered Markov chains. Math. Operat. Res. 20, 182–194.Google Scholar
Malyshev, V. A., and Spieksma, F. M. (1995). Intrinsic convergence rate of countable Markov chains. Markov Proc. Rel. Fields 1, 175–238.Google Scholar
Marshall, A. W., and Shaked, M. (1986). NBU processes with general state space. Math. Operat. Res. 11, 95–109.CrossRefGoogle Scholar
Meyn, S. P., and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.Google Scholar
Meyn, S. P., and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Prob. 4, 981–1011.Google Scholar
Ney, P. (1981). A refinement of the coupling method in renewal theory. Stoch. Proc. Appl. 11, 11–26.Google Scholar
Pitman, J. W. (1974). Uniform rates of convergence for Markov chain transition probabilities. Z. Wahrscheinlich-keitsth. 29, 193–227.Google Scholar
Roberts, G. O., and Polson, N. G. (1994). On the geometric convergence of the Gibbs sampler. J. R. Statist. Soc. Ser. B 56, 377–384.Google Scholar
Roberts, G. O., and Tweedie, R. L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stoch. Proc. Appl. 80, 211–229.Google Scholar
Rogozin, B. A. (1973). An estimate of the remainder term in limit theorems in renewal theory. Theory Prob. Appl. 18, 662–677.Google Scholar
Rosenthal, J. S. (1995). Convergence rates for Markov chains. SIAM Rev. 37, 387–405.Google Scholar
Sengupta, D., Chatterjee, A., and Chakraborty, B. (1995). Reliability bounds and other inequalities for discrete life distributions. Microelectronics Rel. 35, 1473–1478.Google Scholar
Shanthikumar, J. G. (1984). Processes with new better than used first-passage times. Adv. Appl. Prob. 16, 667–686.Google Scholar
Smith, W. L. (1958). Renewal theory and its ramifications. J. R. Statist. Soc. Ser. B 20, 243–302.Google Scholar
Stone, C. (1965). On moment generating functions and renewal theory. Ann. Math. Statist. 36, 1298–1301.Google Scholar
Stone, C., and Wainger, S. (1967). One sided error estimates in renewal theory. J. Anal. Math. 20, 325–352.Google Scholar