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

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).) There is a dearth of particular examples of processes and trends for which the distributions of interest are known exactly. Such examples could give useful experience of the form of distribution to be expected in typical cases, and could serve as material on which to test out approximate methods. The object of the present paper is to provide an example of this kind. One process for which exact results are available in the trend-free case is the Ornstein-Uhlenbeck process, i.e., the stationary Gaussian Markov process X(t) generated by