  The Sparre-Andersen theorem is a remarkable result in one-dimensional random
walk theory concerning the universality of the ubiquitous first-passage-time
distribution. It states that the probability distribution $\rho_n$ of the
number of steps needed for a walker starting at the origin to land on the
positive semi-axes does not depend on the details of the distribution for the
jumps of the walker, provided this distribution is symmetric and continuous,
where in particular $\rho_n \sim n^{-3/2}$ for large number of steps $n$. On
the other hand, there are many physical situations in which the time spent by
the walker in doing one step depends on the length of the step and the interest
concentrates on the time needed for a return, not on the number of steps. Here
we modify the Sparre-Andersen proof to deal with such cases, in rather general
situations in which the time variable correlates with the step variable. As an
example we present a natural process in 2D that shows deviations from normal
scaling are present for the first-passage-time distribution on a semi plane.
