On a limit behaviour of a random walk penalised in the lower half-plane

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.