  We show analogs of the classical arcsine theorem for the occupation time of a
random walk in $(-\infty,0)$ in the case of a small positive drift. To study
the asymptotic behavior of the total time spent in $(-\infty,0)$ we consider
parametrized classes of random walks, where the convergence of the parameter to
zero implies the convergence of the drift to zero. We begin with shift
families, generated by a centered random walk by adding to each step a shift
constant $a>0$ and then letting $a$ tend to zero. Then we study families of
associated distributions. In all cases we arrive at the same limiting
distribution, which is the distribution of the time spent below zero of a
standard Brownian motion with drift 1. For shift families this is explained by
a functional limit theorem. Using fluctuation-theoretic formulas we derive the
generating function of the occupation time in closed form, which provides an
alternative approach.
  In the course also give a new form of the first arcsine law for the Brownian
motion with drift.

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