Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-16T05:20:33.763Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

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

Kartvelishvili, N. V. (1969) Theory of Stochastic Processes in Hydrology and River Runoff Regulation. Israel Program for Scientific Translations, Jerusalem.Google Scholar
Lampard, D. G. (1968) A stochastic process whose successive intervals between events form a first order Markov chain. J. Appl. Prob. 5, 648–668.CrossRefGoogle Scholar
Lukacs, E. and Laha, R. G. (1964) Applications of Characteristic Functions. Griffin, London.Google Scholar
McKay, A. T. (1932) A Bessel function distribution. Biometrika 24, 39–44.CrossRefGoogle Scholar
Phatarfod, R. M. (1971) Some approximate results in renewal and dam theories. J. Austral. Math. Soc. 12, 425–431.CrossRefGoogle Scholar
Phatarfod, R. M. (1976) Some aspects of stochastic reservoir theory. J. Hydrol. 30, 199–217.CrossRefGoogle Scholar