  We consider a random walk $\tilde S$ which has different increment
distributions in positive and negative half-planes. In the upper half-plane the
increments are mean-zero i.i.d. with finite variance. In the lower half-plane
we consider two cases: increments are positive i.i.d. random variables with
either a slowly varying tail or with a finite expectation. For the
distributions with a slowly varying tails, we show that $\{\frac{1}{\sqrt n}
\tilde S(nt)\}$ has no weak limit in $\De$; alternatively, the weak limit is a
reflected Brownian motion.
