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Geomatric ergodicity and quasi-stationarity in discrete-time birth-death processes

Published online by Cambridge University Press:  17 February 2009

Erik A. van Doorn  and
Pauline Schrijner
Erik A. van Doorn
Affiliation:
Faculty of Applied Math., University of Twente, 7500 AE Enschede, The Netherlands.
Pauline Schrijner
Affiliation:
Faculty of Applied Math., University of Twente, 7500 AE Enschede, The Netherlands.
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Abstract

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We study two aspects of discrete-time birth-death processes, the common feature of which is the central role played by the decay parameter of the process. First, conditions for geometric ergodicity and bounds for the decay parameter are obtained. Then the existence and structure of quasi-stationary distributions are discussed. The analyses are based on the spectral representation for the n-step transition probabilities of a birth-death process developed by Karlin and McGregor.

Type
Research Article
Information
The ANZIAM Journal , Volume 37 , Issue 2 , October 1995 , pp. 121 - 144
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Chihara, T. S., An introduction to orthogonal polynomials (Gordon and Breach, New York, 1978).Google Scholar
[2]Chihara, T. S., ‘Spectral properties of orthogonal polynomials on unbounded sets’, Trans. Amer. Math. Soc. 270 (1982) 623–639.CrossRefGoogle Scholar
[3]Derman, C., ‘A solution to a set of fundamental equations in Markov chains’, Proc. Amer. Math. Soc. 5 (1954) 332–334.CrossRefGoogle Scholar
[4]Doom, E. A. van, ‘On oscillation properties and the interval of orthogonality of orthogonal polynomials’, SIAM J. Math. Anal. 15 (1984) 1031–1042.Google Scholar
[5]Doom, E. A. van, ‘Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process’, Adv. Appl. Probab. 17 (1985) 514–530.Google Scholar
[6]Doom, E. A.van, ‘Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices’, J. Approx. Th. 51 (1987) 254–266.Google Scholar
[7]Doom, E. A.van, ‘Quasi-stationary distributions and convergence to quasi-stationarity of birth death processes’, Adv. Appl. Probab. 23 (1991) 683–700.Google Scholar
[8]Doom, E. A.van and Schrijner, P., ‘Random walk polynomials and random walk measures’, J. Comput. Appl. Math. 49 (1993) 289–296.Google Scholar
[9]Doom, E. A.van and Schrijner, P., ‘Ratio limits and limiting conditional distributions for discretetime birth-death processes’, J. Math. Anal. Appl. 190 (1995) 263–284.Google Scholar
[10]Ferrari, P. A., Kesten, H., Martínez, S. and Picco, P., ‘Existence of quasistationary distributions. A renewal dynamical approach’, Ann. Probab., to appear.Google Scholar
[11]Ferrari, P. A., Martínez, S. and Picco, P., ‘Existence of non trivial quasi stationary distributions in the birth and death chain’, Adv. Appl. Probab. 24 (1992) 795–813.CrossRefGoogle Scholar
[12]Harris, T. E., ‘First passage and recurrence distributions’, Trans. Amer. Math. Soc. 73 (1952) 471–486.CrossRefGoogle Scholar
[13]Karlin, S. and McGregor, J. L., ‘The differential equations of birth-and-death processes, and the Stieltjes moment problem’, Trans. Amer. Math. Soc. 85 (1957) 489–546.CrossRefGoogle Scholar
[14]Karlin, S. and McGregor, J. L., ‘The classification of birth and death processes’, Trans. Amer. Math. Soc. 86 (1957), 366–400.CrossRefGoogle Scholar
[15]Karlin, S. and McGregor, J. L., ‘Random walks’, Illinois J. Math. 3 (1959), 66–81.CrossRefGoogle Scholar
[16]Kendall, D. G., ‘Unitary dilations of Markov transition operators and the corresponding integral representation for transition probability matrices’, pp. 139–161, in: Surveys in probability and statistics - The Harold Cramer volume, Grenander, U., ed. (Almquist and Wiksell, Stockholm, 1959).Google Scholar
[17]Kent, J. T. and Longford, N. T., ‘An eigenvalue decomposition for first hitting times in random walks’, Z. Wahrscheinlichkeitstheorie verw. Gebiete 63 (1983) 71–84.CrossRefGoogle Scholar
[18]Kersting, G., ‘Strong ratio limit property and R-recurrence of reversible Markov chains’, Z. Wahr scheinlichkeitstheorie verw. Gebiete 30 (1974) 343–356.CrossRefGoogle Scholar
[19]Kijima, M., ‘On the existence of quasi-stationary distributions in denumerable R-transient Markov chains’, J. Appl. Probab. 29 (1992) 21–36; Correction, 30 (1993) 496.CrossRefGoogle Scholar
[20]Kijima, M., ‘Quasi-stationary distributions of single-server phase-type queues’, Math. Oper. Res. 18 (1993) 423–437.CrossRefGoogle Scholar
[21]Kingman, J. F. C., ‘The exponential decay of Markov transition probabilities’, Proc. London Math Soc. 13 (1963) 337–358.CrossRefGoogle Scholar
[22]Nair, M. G. and Pollett, P. K., ‘On the relationship betweenμ-invariant measures and quasistationary distributions for continuous-time Markov chains’, Adv. Appl. Probab. 25 (1993) 82–102.CrossRefGoogle Scholar
[23]Papangelou, F., ‘Strong ratio limits, R-recurrence and mixing properties of discrete parameter Markov processes’, Z. Wahrscheinlichkeitstheorie verw. Gebrete 8 (1967) 259–297.CrossRefGoogle Scholar
[24]Pollett, P. K., ‘Recent advances in the theory and application of quasistationary distributions’, pp. 477–486, in Stochastic models in engineering, technology and management, eds. Osaki, S. and Murthy, D. N. P., (World Scientific, Singapore, 1993).Google Scholar
[25]Sansigre, G. and Valent, G., ‘A large family of semi-classical polynomials: the perturbed Tchebichev’, J. Comput. Appl. Math, to appear.Google Scholar
[26]Seneta, E. and Vere-Jones, D., ‘On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states’, J. Appl. Probab. 3 (1966) 403–434.CrossRefGoogle Scholar
[27]Shohat, J. A. and Tamarkin, J. D., The problem of moments. Mathematical Surveys No I, rev. edition (American Mathematical Society, Providence, R. I., 1963).Google Scholar
[28]Vere-Jones, D., ‘Geometric ergodicity in denumerable Markov chains’, Quart. J. Math. Oxford (2) 13 (1962) 7–28.CrossRefGoogle Scholar
[29]Whitehurst, T. A., On random walks and orthogonal polynomials. Ph.D. Thesis (Indiana University, Bloomington, 1978).Google Scholar
[30]Whitt, W. W., ‘The renewal-process stationary-excess operator’, J. Appl. Probab. 22 (1985) 156–167.CrossRefGoogle Scholar