  Insight into a number of interesting questions in cosmology can be obtained
from the first crossing distributions of physically motivated barriers by
random walks with correlated steps. We write the first crossing distribution as
a formal series, ordered by the number of times a walk upcrosses the barrier.
Since the fraction of walks with many upcrossings is negligible if the walk has
not taken many steps, the leading order term in this series is the most
relevant for understanding the massive objects of most interest in cosmology.
This first term only requires knowledge of the bivariate distribution of the
walk height and slope, and provides an excellent approximation to the first
crossing distribution for all barriers and smoothing filters of current
interest. We show that this simplicity survives when extending the approach to
the case of non-Gaussian random fields. For non-Gaussian fields which are
obtained by deterministic transformations of a Gaussian, the first crossing
distribution is simply related to that for Gaussian walks crossing a suitably
rescaled barrier. Our analysis shows that this is a useful way to think of the
generic case as well. Although our study is motivated by the possibility that
the primordial fluctuation field was non-Gaussian, our results are general. In
particular, they do not assume the non-Gaussianity is small, so they may be
viewed as the solution to an excursion set analysis of the late-time, nonlinear
fluctuation field rather than the initial one. They are also useful for models
in which the barrier height is determined by quantities other than the initial
density, since most other physically motivated variables (such as the shear)
are usually stochastic and non-Gaussian. We use the Lognormal transformation to
illustrate some of our arguments.
