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Minification processes with discrete marginals

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

V. A. Kalamkar
V. A. Kalamkar*
Affiliation:
The M. S. University of Baroda
*
∗Postal address: Department of Statistics, The M. S. University of Baroda, Vadodara 390 002, India.
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Abstract

We investigate the stationarity of minification processes when the marginal is a discrete distribution. There is a close relationship between the problem considered by Arnold and Isaacson (1976) and the stationarity in minification processes. We give a necessary and sufficient condition for a discrete distribution to be the marginal of a stationary minification process. Members of the Poisson and negative binomial families can be the marginals of stationary minification processes. The geometric minification process is studied in detail, and two characterizations of it based on the structure of the innovation process are given.

Keywords

POISSON DISTRIBUTIONNEGATIVE BINOMIAL DISTRIBUTIONGEOMETRIC DISTRIBUTIONEXPONENTIAL DISTRIBUTIONCHARACTERIZATIONINNOVATION PROCESS

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60G10: Stationary processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 3 , September 1995 , pp. 692 - 706
Copyright
Copyright © Applied Probability Trust 1995 

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References

Adke, S. R. and Balakrishna, N. (1992) Estimation of the mean of some stationary Markov processes. Commun. Statist. Theory Meth. 21, 137–159.CrossRefGoogle Scholar
Alpuim, M. T. (1989) An extremal Markovian sequence. J. Appl. Prob. 26, 219–232.CrossRefGoogle Scholar
Alpuim, M. T. and Athayde, E. (1990) On the stationary distributions of some extremal Markovian sequences. J. Appl. Prob. 27, 291–302.CrossRefGoogle Scholar
Arnold, B. C. and Isaacson, D. (1976) On solutions to and . Z. Wahrschlichkeitsth. verw. Gebiete, 115–119.CrossRefGoogle Scholar
Chernick, M. R., Daley, D. J. and Littlejohn, R. P. (1988) A time reversibility relationship between two Markov chains with stationary exponential distribution. J. Appl. Prob. 25, 418–422.CrossRefGoogle Scholar
Lewis, P. A. W. and Mckenzie, E. (1991) Minification processes and their transformations. J. Appl. Prob. 28, 45–57.CrossRefGoogle Scholar
Littlejohn, R. P. (1992) Discrete minification processes and reversibility. J. Appl. Prob. 29, 82–91.CrossRefGoogle Scholar
Sim, C. H. (1986) Simulation of Weibull and gamma autoregressive stationary processes. Commun. Statist. Simul. 5, 1141–1146.CrossRefGoogle Scholar
Tavares, L. V. (1980) An exponential Markovian stationary process. J. Appl. Prob. 17, 1117–1120.CrossRefGoogle Scholar