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A note on seasonal Markov chains with gamma or gamma-like distributions

Published online by Cambridge University Press:  14 July 2016

E. H. Lloyd  and
S. D. Saleem
E. H. Lloyd*
Affiliation:
University of Lancaster
S. D. Saleem*
Affiliation:
University of Lancaster
*
∗Postal address: Department of Mathematics, University of Lancaster, Bailrigg, Lancaster, U.K.
∗Postal address: Department of Mathematics, University of Lancaster, Bailrigg, Lancaster, U.K.
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Abstract

Weighted sums defined on a Markov chain (MC) are important in applications (e.g. to reservoir storage theory). The rather intractable theory of such sums simplifies to some extent when the transition p.d.f. of the chain {Xt} has a Laplace transform (LT) L(Xt+1; θ |Χ t=x) of the ‘exponential' form H(θ) exp{ – G(θ)x}. An algorithm is derived for the computation of the LT of Σat,Χ t for this class, and for a seasonal generalization of it.

A special case of this desirable exponential type of transition LT for a continuous-state discrete-time MC is identified by comparison with the LT of the Bessel distribution. This is made the basis for a new derivation of a gamma-distributed MC proposed by Lampard (1968).

A seasonal version of this process is developed, valid for any number of seasons.

Reference is made to related chains with three-parameter gamma-like distributions (of the Kritskii–Menkel family) that may be generated from the above by a simple power transformation.

Keywords

MARKOV CHAINSSEASONALITYWEIGHTED SUMS OF MARKOVIAN VARIABLESBESSEL DISTRIBUTIONKRITSKII–MENKEL DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 1 , March 1979 , pp. 117 - 128
Copyright
Copyright © Applied Probability Trust 1979 

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References

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