Hostname: page-component-7dd5485656-bvgqh Total loading time: 0 Render date: 2025-10-29T10:01:27.879Z Has data issue: false hasContentIssue false
  • English
  • Français

Coefficients of ergodicity: structure and applications

Published online by Cambridge University Press:  01 July 2016

E. Seneta
E. Seneta*
Affiliation:
The Australian National University
*
∗ Postal addresss: Department of Statistics, The Australian National University, S.G.S., P.O. Box 4, Canberra A.C.T. 2600, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

The concept of ‘coefficient of ergodicity’, τ(P), for a finite stochastic matrix P, is developed from a standpoint more general and less standard than hitherto, albeit synthesized from ideas in existing literature. Several versions of such a coefficient are studied theoretically and by numerical examples, and usefulness in applications compared from viewpoints which include the degree to which extension to more general matrices is possible. Attention is given to the less familiar spectrum localization property: where λ is any non-unit eigenvalue of P. The essential purpose is exposition and unification, with the aid of simple informal proofs.

Keywords

FINITE STOCHASTIC MATRICESCOEFFICIENTS OF ERGODICITYNON-NEGATIVE MATRICESSPECTRUM LOCALIZATIONGENERALIZED SCRAMBLING MATRICES

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 3 , September 1979 , pp. 576 - 590
Copyright
Copyright © Applied Probability Trust 1979 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Invited paper presented to 8th Conference on Stochastic Processes and their Applications, Canberra, 6–10 July, 1978.

References

1. Alpin, Yu. A. and Gabassov, N. Z. (1976) A remark on the problem of localization of the eigenvalues of real matrices (in Russian). Izv. Vyssh. Uchebn. Zaved. Matematika 11 (174), 98–100.Google Scholar
2. Anthonisse, J. M. and Tijms, H. (1977) Exponential convergence of products of stochastic matrices. J. Math. Anal. Appl. 59, 360–364.Google Scholar
3. Bernstein, S. N. (1946) Teoriya Veroiatnostei, 4th edn. Gostehizdat, Moscow–Leningrad. (Relevant portions are reprinted in [4].) Google Scholar
4. Bernstein, S. N. (1964) Classification of Markov chains and their matrices (in Russian). Sobranie Sochineniy: Tom TV, Teoriya Veroiatnostei i Matematicheskaia Statistika [1911–1946], pp. 455–483. Nauka, Moscow.Google Scholar
5. Birkhoff, G. (1967) Lattice Theory, 3rd edn. Amer. Math. Soc. Colloq. Publicns. Vol XXV, Providence, R.I. Google Scholar
6. Blagoveshchenskii, Yu. N. (1960) On ergodicity for a scheme of series of Markov chains with a finite number of states and discrete time (in Russian) Izv. Akad. Nauk Uzbek. SSR. Ser. fiz.-mat. 4, (3), 7–15.Google Scholar
7. Brauer, A. (1952) Limits for the characteristic roots of a matrix IV: Applications to stochastic matrices. Duke Math. J. 19, 75–91.Google Scholar
8. Chatterjee, S. and Seneta, E. (1977) Towards consensus: some convergence theorems on repeated averaging. J. Appl. Prob. 14, 89–97.Google Scholar
9. Cohn, H. (1976) Finite non-homogeneous Markov chains: asymptotic behaviour. Adv. Appl. Prob. 8, 502–516.Google Scholar
10. Dobrushin, R. L. (1956) Central limit theorem for non-stationary Markov chains, I, II. Theory Prob. Appl. 1, 65–80, 329–383. (English translation.) Google Scholar
11. Golubitsky, M., Keeler, E. B. and Rothschild, M. (1975) Convergence of the age structure; applications of the projective metric. Theoret. Popn Biol. 7, 84–93.Google Scholar
12. Hajnal, J. (1958) Weak ergodicity in non-homogeneous Markov chains. Proc. Camb. Phil. Soc. 54, 233–246.Google Scholar
13. Hajnal, J. (1976) On products of non-negative matrices. Math. Proc. Camb. Phil. Soc. 79, 521–530.Google Scholar
14. Iosifescu, M. (1977) Lanturi Markov Finite şi Aplicaţii. ed. Tehniča, Bucharest.Google Scholar
15. Isaacson, D. L. and Madsen, R. W. (1976) Markov Chains. Wiley, New York.Google Scholar
16. Kingman, J. F. C. (1975) Geometrical aspects of the theory of non-homogeneous Markov chains. Math. Proc. Camb. Phil. Soc. 77, 171–183.Google Scholar
17. Kolmogorov, A. N. (1931) Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104, 415–458.Google Scholar
18. Krysanov, A. I., Kuzin, L. T., and Letunov, Yu. P. (1971) Some asymptotic estimates of the convergence of discrete Markov chains. Engineering Cybernetics 9, 1095–1101. (English translation.)Google Scholar
19. Paz, A. (1965) Definite and quasidefinite sets of stochastic matrices. Proc. Amer. Math. Soc. 16, 634–641.Google Scholar
20. Paz, A. (1971) Introduction to Probabilistic Automata. Academic Press, New York.Google Scholar
21. Pykh, Yu. A. (1973) The characteristic numbers of stochastic matrices (in Russian). Dokl. Akad. Nauk S.S.S.R. 211, 1299–1301.Google Scholar
22. Sarymsakov, T. A. (1954) Osnovi Teorii Processov Markova. G.I.T.-T.L., Moscow.Google Scholar
23. Sarymsakov, T. A. (1956) On the theory of inhomogeneous Markov chains (in Russian). Dokl. Akad. Nauk Uzbek S.S.R. 8, 3–7.Google Scholar
24. Sarymsakov, T. A. (1958) On inhomogeneous Markov chains (in Russian). Dokl. Akad. Nauk S.S.S.R. 120, 465–467.Google Scholar
25. Sarymsakov, T. A. (1961) Inhomogeneous Markov chains (in Russian). Teor. Veroiat. Primenen. 6, 194–201.Google Scholar
26. Seneta, E. (1973) On the historical development of the theory of finite inhomogeneous Markov chains. Proc. Camb. Phil. Soc. 74, 507–513 Google Scholar
27. Seneta, E. (1973) On strong ergodicity of inhomogeneous products of finite stochastic matrices. Studia Math. 46, 241–247.Google Scholar
28. Seneta, E. (1973) Non-negative Matrices. Allen and Unwin, London.Google Scholar
29. Seneta, E. (1974) [Review of [21], #11167.] Math. Rev. 48, No. 6, p. 1296.Google Scholar
30. Wolfowitz, J. (1963) Products of indecomposable, aperiodic, stochastic matrices. Proc. Amer. Math. Soc. 14, 733–737.CrossRefGoogle Scholar
31. Zenger, C. (1972) A comparison of some bounds for the non-trivial eigenvalues of stochastic matrices. Numerische Math. 19, 209–211.Google Scholar