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Markoff chains with an enumerable number of states and a class of cascade processes

Published online by Cambridge University Press:  24 October 2008

F. G. Foster
F. G. Foster
Affiliation:
The Queen's UniversityBelfast
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Abstract

In § 1 a Markoff chain is defined, and a theorem of Kolmogoroff relating to its asymptotic behaviour is stated. Its stable distributions are examined in § 2 and some further results are obtained. These are then applied in §§ 3, 4 to a certain generalization of the cascade process, regarded as a Markoff chain with a special kind of matrix.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 1 , January 1951 , pp. 77 - 85
Copyright
Copyright © Cambridge Philosophical Society 1951

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References

REFERENCES

(1)Fréchet, M.Recherches théoriques modernes. Livre ii. Traité du calcul des probabilités et de ses applications, tome 1 (Paris, 1938), fasc. 3.Google Scholar
(2)Yosida, K. and Kakutani, S.Markoff process with an enumerable infinite number of possible states. Japanese J. Math. 16 (1939), 47–55.CrossRefGoogle Scholar
(3)Yosida, K. and Kakutani, S.Operator-theoretical treatment of Markoff's process and mean ergodic theorem. Ann. Math. 42 (1941), 188–288.CrossRefGoogle Scholar
(4)Doob, J. L.Topics in the theory of Markoff chains. Trans. American Math. Soc. 52 (1942), 37–64.CrossRefGoogle Scholar
(5)Doob, J. L.Markoff chains—denumerable case. Trans. American Math. Soc. 58 (1945), 455–73.Google Scholar
(6)Doob, J. L.Asymptotic properties of Markoff transition probabilities. Trans. American Math. Soc. 63 (1948), 393–421.Google Scholar
(7)Riesz, F.Über lineare Functionalgleichungen. Acta Math. 41 (1918), 71–98.CrossRefGoogle Scholar
(8)Good, I. J.The number of individuals in a cascade process. Proc. Cambridge Phil. Soc. 45 (1949), 360–3.CrossRefGoogle Scholar
(9)Otter, R.The multiplicative process. Ann. Math. Statist. 20 (1949), 206–24.CrossRefGoogle Scholar