Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-15T01:59:17.418Z Has data issue: false hasContentIssue false
  • English
  • Français

A central limit theorem for processes defined on a finite Markov chain

Published online by Cambridge University Press:  24 October 2008

J. Keilson  and
D. M. G. Wishart
J. Keilson
Affiliation:
University of Birmingham
D. M. G. Wishart
Affiliation:
University of Birmingham
Get access Rights & Permissions [Opens in a new window]

Extract

We shall be concerned in this paper with a class of temporally homogeneous Markov processes, {R(t), X(t)}, in discrete or continuous time taking values in the space

The marginal process {X(t)} in discrete time is, in the terminology of Miller (10), a sequence of random variables defined on a finite Markov chain. Probability measures associated with these processes are vectors of the form

where

We shall call a vector of the form of (0·2) a vector distribution.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 3 , July 1964 , pp. 547 - 567
Copyright
Copyright © Cambridge Philosophical Society 1964

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.)

References

REFERENCES

(1) Courant, R. Differential and integral calculus, vol. 2 (Blackie, 1936).Google Scholar
(2) Cramér, H. Mathematical methods of statistics (Princeton, 1946).Google Scholar
(3) Daniels, H. E. Saddlepoint approximations and statistics. Ann. Math. Statist. 25 (1954), 631–650.CrossRefGoogle Scholar
(4) Debreu, G. and Herstein, I. N. Non-negative square matrices. Econometrica, 21 (1953), 597–607.CrossRefGoogle Scholar
(5) Friedman, B. Principles and techniques of applied mathematics (Wiley, 1956).Google Scholar
(6) Frobenius, G. Über Matrizen aus positiven Elementen, I. S.-B. Akad. Wiss., Berlin (1908), 471–476.Google Scholar
(7) Frobenius, G. Über Matrizen aus nicht negativen Elementen. S.-B. Akad. Wiss., Berlin (1912), 456–477.Google Scholar
(8) Keilson, J. The first passage time density for homogeneous skip-free walks on the continuum. Ann. Math. Statist. 34 (1963), 1003–1011.CrossRefGoogle Scholar
(9) Kingman, J. F. C. A convexity property of positive matrices. Quart. J. Math. Oxford Ser. (2), 12 (1961), 283–284.CrossRefGoogle Scholar
(10) Miller, H. D. A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32 (1961), 1260–1270.CrossRefGoogle Scholar
(11) Neveu, J. Une généralisation des processus à accroissements positifs indépendants. Abh. Math. Sem. Univ. Hamburg, 25 (1961), 36–61.CrossRefGoogle Scholar
(12) Romanovsky, V. Recherches sur les chaines de Markoff. Acta Math. 66 (1936), 147–251.CrossRefGoogle Scholar
(13) Van Der Waerden, B. L. Modern algebra, vol. 1 (Ungar, 1949).Google Scholar
(14) Volkov, I. S. On the distribution of sums of random variables defined on a homogeneous Markov chain with a finite number of states. Teor. Veroyatnost. i Primenen. 3 (1958), 413–429. (In Russian.)Google Scholar