Approximating excursion distributions using regenerative processes

The excursion distribution of stochastic processes is important in many applications and is an important object in probability theory. The excursion distributions are defined as the time between a u-level up-crossing and the following down-crossing of the same level. While the notion of the excursion distribution might be intuitive, the general problem of describing it is not. Finding the excursion distribution continues to pose a problem and has been doing so during the last century. Due to the need to describe the excursion distribution in applications, several approximation methods have been developed.

One such method is the independent interval approximation. This method uses a stationary alternating renewal process to approximate the clipped process. The clipped process is computed by taking the sign of the process we are interested in. Which is usually a sufficiently smooth stationary Gaussian process. Then the interval distribution is found by matching the characteristics of the clipped processes to those of the stationary alternating renewal process.

In this thesis, a new approximation method is introduced, which is similar to the independent interval approximation. However, this method is based on the Slepian process, which models the behavior of a smooth stationary Gaussian process at the moment of an up-crossing of level u. The Slepian process is not stationary. Therefore, a non-stationary alternating renewal process is used to approximate the clipped Slepian process. This is the main idea behind the Slepian-based independent interval approximation. Paper II introduces this for zero-level excursion, and in Paper IV, it is extended to more general crossing levels. These two papers treat one of the two major topics within this thesis.

The second major topic of this thesis concerns the characteristics of alternating renewal processes. In particular, when the interval distributions can be recovered from characterizing functions such as the covariance or expected value functions. In Paper I, the symmetric case is treated, and it is shown that this is possible under monotonicity conditions on the expected value function. If this condition is satisfied, the interval distribution will be geometrically divisible. The asymmetric case is treated in Paper III, and similar conditions are derived.

The results concerning the recoverability of the interval distribution in Papers I and III have practical implications for the Slepian-based independent interval approximation. They ensure that the obtained approximation of the excursion distribution is a valid probability distribution, which is not obvious for the ordinary independent interval approximation.