Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T08:41:14.366Z Has data issue: false hasContentIssue false
  • English
  • Français

The Geometry of Finite Markov Chains

Published online by Cambridge University Press:  20 November 2018

N. Pullman
N. Pullman*
Affiliation:
McGill University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this paper is to present a geometric theorem which provides a proof of a fundamental theorem of finite Markov chains.

The theorem, stated in matrix theoretic terms, concerns the asymptotic behaviour of the powers of an n by n stochastic matrix, that is, a matrix of non-negative entries each of whose row sums is 1. The matrix might arise from a repeated physical process which goes from one of n possible states to another at each iteration and whose probability of going to a state depends only on the state it is in at present and not on its more distant history.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 8 , Issue 3 , April 1965 , pp. 345 - 358
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Stochastic Processes, Doob, J., John Wiley and Sons, New York, 1953.Google Scholar
2. An Introduction to Probability Theory and its Applications, Volume one, Feller, W., 2nd edition, John Wiley and Sons, New York, 1950.Google Scholar
3. Markov Chains with Stationary Transition Probabilities, Chung, Kai Lai, Springer, Berlin 1960.CrossRefGoogle Scholar
4. Finite Markov Chains, Kemeny, J. G. and Snell, J. Laurie, D. van Nostrand Co., New Jersey 1960.Google Scholar
5. Applications of the Theory of Matrices, Gantmacher, F. R., Interscience, New York 1959.Google Scholar
6. Convexity, Eggleston, H. G., Cambridge Tracts in Mathematics and Mathematical Physics No. 47, Cambridge University Press, 1958.CrossRefGoogle Scholar
7. Pullman, N., Infinite Products of Substochastic Matrices, Pacific Journal (to appear).Google Scholar