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Cumulants of some distributions arising from a two-state Markoff chain

Published online by Cambridge University Press:  24 October 2008

P. V. Krishna Iyer  and
N. S. Shakuntala
P. V. Krishna Iyer
Affiliation:
Defence Science LaboratoryNew DelhiIndia
N. S. Shakuntala
Affiliation:
Defence Science LaboratoryNew DelhiIndia
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Extract

The asymptotic values of the variances and covariances for the distribution of the transition numbers AA, AB, BA and BB for a two-state Markoff chain have been given by Bartlett (l). Whittle (3) obtained the probability distribution and their factorial moments for the k-state chain under the restriction that the first and the last observations belong to the rth and the sth state. But the expressions for the factorial moments according to Whittle himself are of little immediate use. He obtains useful results for the low order factorial moments by approximating in a grosser fashion, and the evaluation of the cumulants from these factorial moments is rather cumbersome. The object of this note is to give the first four cumulants and product cumulants which are exact to the order nr (p11 – p21)n−r where r is the order of the cumulant, for the distribution of the transition numbers of a two-state Markoff chain. They have been calculated by using the method developed by Iyer and Kapur(2). This method can equally be used for evaluating the asymptotic values of the cumulants for the k-state also. This is done by differentiating directly the determinantal characteristic equation.

Type
Research Notes
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 55 , Issue 3 , July 1959 , pp. 273 - 276
Copyright
Copyright © Cambridge Philosophical Society 1959

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References

REFERENCES

(1)Bartlett, M. S.Proc. Camb. Phil. Soc. 47 (1951), 84–95.Google Scholar
(2)Krishna, Iyer P. V. and Kapur, M. N.Biometrika, 41 (1954), 553–4.Google Scholar
(3)Whittle, P.J. Roy. Statist. Soc. B, 17 (1955), 235–42.Google Scholar