  This work is to popularize the method of computing the distribution of the
excursion times for a Gaussian process that involves extended and multivariate
Rice's formula. The approach was used in numerical implementations of the
high-dimensional integration routine and in earlier work it was shown that the
computations are more effective and thus more precise than those based on Rice
expansions.
  The joint distribution of excursion times is related to the distribution of
the number of level crossings, a problem that can be attacked via the Rice
series expansion, based on the moments of the number of crossings. Another
point of attack is the "Independent Interval Approximation" intensively studied
for the persistence of physical systems. It treats the lengths of successive
crossing intervals as statistically independent. A renewal type argument leads
to an expression that provides the approximate interval distribution via its
Laplace transform.
  However, independence is not valid in typical situations. Even if it leads to
acceptable results for the persistency exponent, rigorous assessment of the
approximation error is not available. Moreover, we show that the IIA approach
cannot deliver properly defined probability distributions and thus the method
is limited to persistence studies.
  This paper presents an alternative approach that is both more general, more
accurate, and relatively unknown. It is based on exact expressions for the
probability density for one and for two successive excursion lengths. The
numerical routine RIND computes the densities using recent advances in
scientific computing and is easily accessible via a simple Matlab interface.
The result solves the problem of two-step excursion dependence for a general
stationary differentiable Gaussian process. The work offers also some
analytical results that explain the effectiveness of the implemented method.
