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A process with chain dependent growth rate

Published online by Cambridge University Press:  14 July 2016

Julian Keilson  and
S. Subba Rao
Julian Keilson
Affiliation:
University of Rochester, Rochester, New York
S. Subba Rao
Affiliation:
University of Rochester, Rochester, New York
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Extract

Additive processes on finite Markov chains have been investigated by Miller ([8], [9]), Keilson and Wishart ([2], [3], [4]) and by Fukushima and Hitsuda [1]. These papers study a two-dimensional Markov Process {X(t), R(t)} whose state space is R1 × {1, 2, ···, R} characterized by the following properties:

  1. (i) R(t) is an irreducible Markov chain on states 1,2, …,R governed by atransition probability matrix Bo = {brs}.

  2. (ii) X(t) is a sum of random increments dependent on the chain, i.e., if the ith transition takes the chain from state r to state s, then the increment has the distribution

  3. (iii) Nt, is t in discrete time while in the continuous time case Nt, might be an independent Poisson process.

Type
Research Papers
Information
Journal of Applied Probability , Volume 7 , Issue 3 , December 1970 , pp. 699 - 711
Copyright
Copyright © Applied Probability Trust 1970 

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References

[1] Fukushima, M. and Hitsuda, M. (1967) On a class of Markov processes taking values on lines and the central limit theorem. Nagoya Math. J. 30, 47–56.CrossRefGoogle Scholar
[2] Keilson, J. and Wishart, D. M. G. (1964) A central limit theorem for processes defined on a finite Markov chain. Proc. Camb. Phil. Soc. 60, 547–567.CrossRefGoogle Scholar
[3] Keilson, J. and Wishart, D. M. G. (1965) Boundary problems for additive processes defined on a finite Markov chain. Proc. Camb. Phil. Soc. 61, 173–190.CrossRefGoogle Scholar
[4] Keilson, J. and Wishart, D. M. G. (1967) Addenda to processes defined on a finite Markov chain. Proc. Camb. Phil. Soc. 63, 187–193.CrossRefGoogle Scholar
[5] Keilson, J. (1965) A review of transient behavior in regular diffusion and birth-death processes–Part II. J. Appl. Prob. 2, 405–428.CrossRefGoogle Scholar
[6] Keilson, J. (1969) An intermittent channel with finite storage. Opsearch (India) 6, 109–117.Google Scholar
[7] Kemeny, J. G. and Snell, J. L. (1960) Finite Markov Chains. Van Nostrand, Princeton, N.J.Google Scholar
[8] Miller, H. D. (1961) A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32, 1260–1270.CrossRefGoogle Scholar
[9] Miller, H. D. (1962) Absorption probabilities for sums of random variables defined on a finite Markov chain. Proc. Camb. Phil. Soc. 58, 286–298.CrossRefGoogle Scholar
[10] Miller, R. G. (1963) Continuous time stochastic storage processes with random linear inputs and outputs. J. Math. Mech. 12, 275–291.Google Scholar