Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T02:26:41.141Z Has data issue: false hasContentIssue false
  • English
  • Français

Addenda to processes defined on a finite Markov chain

Published online by Cambridge University Press:  24 October 2008

Julian Keilson  and
David M. G. Wishart
Get access Rights & Permissions [Opens in a new window]

Extract

Introduction. Two previous papers (1,2) have dealt with additive processes defined on finite Markov chains. Such a process in discrete time may be treated as a bivariate Markov process {R(k), X(k)}. The process R(k) is an irreducible Markov chain on states r = 1, 2, …, R governed by a stochastic transition matrix B0 with components brs. The marginal process X(k) ‘defined’ on the chain R(k) is a sum of random increments ξ(i) dependent on the chain, i.e. if the ith transition takes the chain from state r to state s, ξ,(i) is chosen from a distribution function Drs(x) indexed by r and s. The distribution of the bivariate process may be represented by a vector F(x, k) with components

These are generated recursively by the relation

where the increment matrix distribution B(x) has components brsDrs(x). We denote the nth moment of B(x) by Bn = ∫xndB(x), so that B0 = B(∞).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 1 , January 1967 , pp. 187 - 193
Copyright
Copyright © Cambridge Philosophical Society 1967

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)Keilson, J. and Wishart, D. M. G.A central limit theorem for processes defined on a finite Markov chain. Proc. Cambridge Philos. Soc. 60 (1964), 547–567.CrossRefGoogle Scholar
(2)Keilson, J. and Wishart, D. M. G.Boundary problems for additive processes defined on a finite Markov chain. Proc. Cambridge Philos. Soc. 61 (1965), 173–190.CrossRefGoogle Scholar
(3)Kemeny, J. G. and Snell, J. L.Finite Markov chains (Van Nostrand, 1960).Google Scholar
(4)Keilson, J.Green's function methods in probability theory (Griffin, 1965).Google Scholar
(5)Spitzer, F.Principles of random walk (Van Nostrand, 1964).CrossRefGoogle Scholar
(6)Debreu, G. and Herstein, I. N.Non-negative square matrices, Econometrica 21 (1953), 597–607.CrossRefGoogle Scholar