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Finite approximations to infinite non-negative matrices

Published online by Cambridge University Press:  24 October 2008

E. Seneta
E. Seneta
Affiliation:
Australian National University
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Extract

In applying the theory of infinite Markov chains to practical examples, it is important to know how the ergodic properties defined by the infinite stochastic or substochastic matrix under consideration are related to those of the n × n (n = 1, 2, 3, …) truncated corner sub-matrices. In particular, it is of interest whether the relevant eigenvalues and eigenvectors of the truncated matrices in some sense approximate to corresponding quantities for the infinite matrix as n → ∞.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 4 , October 1967 , pp. 983 - 992
Copyright
Copyright © Cambridge Philosophical Society 1967

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References

REFERENCES

(1)Beckmann, M., McGuire, C. B. and Winsten, C.Studies in the Economics of Transportation. Cowles Commission for Research in Economics (Yale University Press, 1956).Google Scholar
(2)Kac, M.Random walk and the theory of Brownian motion. Amer. Math. Monthly 54 (1947), 369–391.Google Scholar
(3)Ledermann, W. and Reuter, G. E. H.Spectral theory for the differential equations of simple birth-and-death processes. Philos. Trans. Roy. Soc. London, Ser. A 246 (1954), 321–369.Google Scholar
(4)Mandl, P.An elementary proof of the ergodic property of birth-and-death processes. (In Czech.) Časopis Pěst. Mat. 89 (1964), 354–358.Google Scholar
(5)Pruitt, W. E.Eigenvalues of non-negative matrices. Ann. Math. Statist. 35 (1964), 1797–1800.Google Scholar
(6)Reuter, G. E. H. and Ledermann, W.On the differential equations for the transition probabilities of Markov processes with enumerably many states. Proc. Cambridge Philos. Soc. 49 (1953), 247–262.Google Scholar
(7)Sarymsakov, T. A.Elements of the theory of Markov processes (in Russian) (G.I.T.T.L. Moscow, 1954).Google Scholar
(8)Seneta, E. and Vere-Jones, D.On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3 (1966), 403–434.Google Scholar
(9)Šidak, Z.Eigenvalues of operators in l-spaces in denumerable Markov chains. Czechoslovak Math. J. 14 (89) (1964), 438–443.CrossRefGoogle Scholar
(10)Takács, L.Stochastic processes (Methuen, London (1960)).Google Scholar
(11)Vere-Jones, D.Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford Ser. 13 (1962), 7–28.Google Scholar
(12)Vere-Jones, D.Ergodic properties of non-negative matrices. I. Pacific J. Math. (to appear).Google Scholar