Abstract
In 1968 Davidson posed the following tantilising problem. Suppose p(at) denotes for t ≥ 0 a p-function or transition function for some state of a Markov chain (with p(0) = 1). For any given t > 0, put p(t) = M. How small is m = inf{p(s): s ≤ t}? In other words, what pairs (m, M) can occur?
The problem remains unsolved but substantial contributions to the eventual solution have been made by V. M. Joshi. This paper reviews Joshi’s work and outlines the current state of play.
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References
Blackwell, D., and D. Freedman (1968), “On the local behavior of Markov transition probabilities”. Annals of Mathematical Statistics 39, 2123–2127.
Bloomfield, P. (1971), “Lower bounds for renewal sequences and p-functions”. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 19 271–273.
Cornish, A. G. (1972), unpublished manuscript.
Cornish, A. G. (1973), “Bounds for p-functions”. Bulletin of the London Mathematical Society 5, 169–175.
Davidson, R. (1968), “Arithmetic and other properties of certain delphic semigroups”. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 10, 120–172.
Griffeath, D. (1973), “The maximal oscillation problem for regenerative phenomena”. Annals of Probability 1, 405–416.
Griffeath, D. (1974), “Inequalities for oscillating p-functions”. Bulletin of the London Mathematical Society 6, 51–56.
Griffeath, D. (1976), “On p-functions with exponential start”. Journal of the London Mathematical Society 14, 445–450.
Joshi, V. M. (1975), “A new bound for standard p-functions”. Annals of Probability 3, 346–352.
Joshi, V. M. (1977a), “An improved upper bound for standard p-functions”. Annals of Probability 5, 999–1003.
Joshi, V. M. (1977b), “An upper bound for exponentially starting standard p-functions”. Sankhya, Series A 39 334–340.
Joshi, V. M. (1981), “A generalization of Davidson’s upper bound for standard jump p-functions”. Journal of the Indian Statistical Association 19, 69–76.
Kendall, D. G., and E. F. Harding (eds.) (1973), Stochastic Analysis. New York: Wiley and Sons.
Kingman, J. F. C. (1964), “The stochastic theory of regenerative events”. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 2, 180–224.
Kingman, J. F. C. (1972), Regenerative Phenomena. New York: Wiley and Sons.
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© 1987 D. Reidel Publishing Company, Dordrecht, Holland
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Cornish, A. (1987). V. M. Joshi and the Markov Oscillation Problem. In: MacNeill, I.B., Umphrey, G.J., Bellhouse, D.R., Kulperger, R.J. (eds) Advances in the Statistical Sciences: Applied Probability, Stochastic Processes, and Sampling Theory. The University of Western Ontario Series in Philosophy of Science, vol 34. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4786-3_1
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DOI: https://doi.org/10.1007/978-94-009-4786-3_1
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