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The minimum of a stationary Markov process superimposed on a U-shaped trend

Published online by Cambridge University Press:  14 July 2016

H.E. Daniels
H.E. Daniels*
Affiliation:
University of Birmingham
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Extract

1. This paper was motivated by some questions of Barnett and Lewis (1967) concerning extreme winter temperatures. The temperature during the winter can be hopefully regarded as generated by a stationary Gaussian process superimposed on a locally U-shaped trend. One is interested in statistical properties of the minimum of sample paths from such a process, and of their excursions below a given level. Equivalently one can consider paths from a stationary process crossing a curved boundary of the same form. Problems of this type are discussed by Cramer and Leadbetter (1967), extensively in the trend-free case and in less detail when a trend is present, following the method initiated by Rice (1945). While results on moments are easy to obtain, explicit results for the actual probability distributions are not usually available. However, in the important case when the level of values of interest is far below the mean, the asymptotic independence of up-crossing times makes it possible to derive simple approximate distributions. (See Cramer and Leadbetter (1967) page 256, Keilson (1966).)

Type
Research Papers
Information
Journal of Applied Probability , Volume 6 , Issue 2 , August 1969 , pp. 399 - 408
Copyright
Copyright © Sheffield: Applied Probability Trust 

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References

Anderson, T. W. and Darling, D. A. (1952) Asymptotic theory of certain goodness of fit criteria based on stochastic processes. Ann. Math. Statist. 23, 193–212.Google Scholar
Barnett, V. D. and Lewis, T. (1967) A study of low temperature probabilities in the context of an industrial problem. J. R. Statist. Soc. A 130, 177–206.Google Scholar
Cramer, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Darling, D. A. and Siegert, J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.Google Scholar
Doob, J. L. (1942) The Brownian motion and stochastic equations. Ann. Math. 2, 351–369.Google Scholar
Keilson, J. (1964) A review of transient behaviour in regular diffusion and birth-death processes. J. Appl. Prob. 1, 247–266.Google Scholar
Keilson, J. (1966) A technique for discussing the passage time distribution for stable systems. J. R. Statist. Soc. B 3, 477–486.Google Scholar
Rice, S. O. (1945) Mathematical analysis of random noise. Bell System Tech. J. 24, 109–157.Google Scholar
Shepp, L. A. (1967) A first passage problem for the Wiener processes. Ann. Math. Statist. 38, 1912–1914.Google Scholar