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The spectral density of a Markov process

Published online by Cambridge University Press:  14 July 2016

B. D. Craven
B. D. Craven*
Affiliation:
University of Melbourne
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Abstract

For a Markov process in discrete time, having finite covariances, both the transient behaviour and the correlation structure may be summed up in a single generating function, here called the key function. If the process is stationary, and the key function possesses a suitable analytic extension, then the process possesses a continuous spectral density, which can be calculated from the key function. A converse result, and some results for the non-stationary process, are also obtained. The calculation is discussed in more detail for the waiting-time process in the GI/G/1 queue.

Keywords

SPECTRAL DENSITYDISCRETE-TIME MARKOV PROCESSSTATIONARY QUEUECORRELATION STRUCTURE
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 3 , September 1973 , pp. 520 - 527
Copyright
Copyright © Applied Probability Trust 1973 

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References

[1] Craven, B. D. (1965) Serial dependence of a Markov process. J. Austral Math. Soc. 5, 299–314.10.1017/S1446788700027737Google Scholar
[2] Craven, B. D. (1969) Asymptotic correlation in a queue. J. Appl. Prob. 6, 573–583.10.2307/3212103Google Scholar
[3] Craven, B. D. (1973) Perturbed Markov processes. (In preparation).Google Scholar
[4] Daley, D. J. (1969) Integral representations of transition probabilities and serial covariances of certain Markov chains. J. Appl. Prob. 6, 648–659.10.2307/3212109Google Scholar
[5] Hobson, E. W. (1926) The Theory of Functions of a Real Variable and the Theory of Fourier's Series. Volume II. Cambridge University Press.Google Scholar