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On Markov chains with an enumerable infinity of states

Published online by Cambridge University Press:  24 October 2008

F. G. Foster
F. G. Foster
Affiliation:
Magdalen CollegeOxford
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Extract

Consider the infinite system of linear ‘inequations’,

in which the coefficients pij have the two properties

We define as properly divergent (p.d.) a solution, {xi} (i = 0, 1, 2, …), such that

and such that |xi| < ∞ for i < ∞. Since {k + xi} is also a solution for any non-negative constant k, there is no loss in generality in assuming (as we always shall) in place of this last condition that

We are concerned with conditions for the existence of p.d. solutions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 4 , October 1952 , pp. 587 - 591
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

REFERENCES

(1)Feller, W.An introduction to probability theory and its applications, vol. 1 (New York, 1950).Google Scholar
(2)Foster, F. G.Markoff chains with an enumerable number of states and a class of cascade processes. Proc. Camb. phil. Soc. 47 (1951), 77–85.CrossRefGoogle Scholar
(3)Kendall, D. G.On non-dissipative Markoff chains with an enumerable infinity of states. Proc. Camb. phil. Soc. 47 (1951), 633–4.CrossRefGoogle Scholar
(4)Knopp, K.Theory and application of infinite series (London, 1928).Google Scholar