Geometric renewal convergence rates from hazard rates

Journal of Applied Probability

Volumn 38, Issue 1, 2001, Pages 180-194

  Berenhaut, Kenneth S ^a   Lund, Robert ^a  

^a UNIVERSITY OF GEORGIA   (United States)

Author keywords

Geometric convergence; Hazard rates; Increasing hazard rate; Markov chains; New better than used; Power series; Renewal sequence

Indexed keywords

EID: 0035533783     PISSN: 00219002     EISSN: None     Source Type: Journal    
DOI: 10.1239/jap/996986652     Document Type: Article

References (33)

1. BERENHAUT, K. S. AND LUND, R. B. (1999). Renewal convergence rates for DHR and NWU lifetimes. Submitted.
2. BERENHAUT, K. S. AND LUND, R. B. (2000). Tech. Rept 2000-13, Department of Statistics, University of Georgia.
3. BERBEE, H. (1987). Convergence rates in the strong law for bounded mixing sequences. Prob. Theory Rel. Fields. 74, 255-270.
4. BROWN, M. (1980). Bounds, inequalities, and monotonicity properties for some specialized renewal processes. Ann. Prob. 8, 227-240.
5. BRYSON, M. C. AND SIDDIQUI, M. M. (1969). Some criteria for aging. J. Amer. Statist. Assoc. 64, 1472-1483.
6. 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.
7. FELLER, W. (1968). An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. John Wiley, New York.
8. GELFOND, A. O. (1964). An estimate for the remainder term in the limit theorem for recurrent events. Theory Prob. Appl. 9, 299-303.
9. GRUBEL, R. (1983). Functions of discrete probability measures: rates of convergence in the renewal theorem. Z. Wahrscheinlichkeitsth. 64, 341-357.
10. HANSEN, B. G. AND FRENK, J. B. G. (1991). Some monotonicity properties of the delayed renewal function. J. Appl. Prob. 28, 811-821.
11. HEATHCOTE, C. R. (1967). Complete exponential convergence and related topics. J. Appl. Prob. 4, 1-40.
12. KENDALL, D. G. (1959). Unitary dilations of Markov transition operators and the corresponding integral representations for transition probability matrices. In Probability and Statistics, ed. U. Grenander. John Wiley, New York, pp. 139-161.
13. KIJIMA, M. (1997). Markov Processes for Stochastic Modeling. Chapman and Hall, London.
14. KONSTANTOPOULOS, T. AND LAST, G. (1999). On the use of Lyapunov function methods in renewal theory. Stoch. Proc. Appl. 79, 165-178.
15. LIGGETT, T. (1989). Total positivity and renewal theory. In Probability, Statistics, and Mathematics. eds. T. W. Anderson, K. B. Athreya and D. L. Iglehart. Academic Press, Boston, pp. 141-162.
16. LINDVALL, T. (1979). On coupling of discrete renewal processes. Z. Wahrscheinlichkeitsth. 48, 57-70.
17. LINDVALL, T. (1992). Lectures on the Coupling Method. John Wiley, New York.
18. LUND, R. B. AND TWEEDIE, R. L. (1996). Geometric convergence rates for stochastically ordered Markov chains. Math. Operat. Res. 20, 182-194.
19. MALYSHEV, V. A. AND SPIEKSMA, F. M. (1995). Intrinsic convergence rate of countable Markov chains. Markov Proc. Rel. Fields 1, 175-238.
20. MARSHALL, A. W. AND SHAKED, M. (1986). NBU processes with general state space. Math. Operat. Res. 11, 95-109.
21. MEYN, S. P. AND TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.
22. MEYN, S. P. AND TWEEDIE, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Prob. 4, 981-1011.
23. NEY, P. (1981). A refinement of the coupling method in renewal theory. Stoch. Proc. Appl. 11, 11-26.
24. PITMAN, J. W. (1974). Uniform rates of convergence for Markov chain transition probabilities. Z. Wahrscheinlichkeitsth. 29, 193-227.
25. 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.
26. ROBERTS, G. O. AND TWEEDIE, R. L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stoch. Proc. Appl. 80, 211-229.
27. ROGOZIN, B. A. (1973). An estimate of the remainder term in limit theorems in renewal theory. Theory Prob. Appl. 18, 662-677.
28. ROSENTHAL, J. S. (1995). Convergence rates for Markov chains. SIAM Rev. 37, 387-405.
29. SENGUPTA, D., CHATTERJEE, A. AND CHAKRABORTY, B. (1995). Reliability bounds and other inequalities for discrete life distributions. Microelectronics Rel. 35, 1473-1478.
30. SHANTHIKUMAR, J. G. (1984). Processes with new better than used first-passage times. Adv. Appl. Prob. 16, 667-686.
31. SMITH, W. L. (1958). Renewal theory and its ramifications. J. R. Statist. Soc. Ser. B 20, 243-302.
32. STONE, C. (1965). On moment generating functions and renewal theory. Ann. Math. Statist. 36, 1298-1301.
33. STONE, C. AND WAINGER, S. (1967). One sided error estimates in renewal theory. J. Anal. Math. 20, 325-352.