  We consider a one dimensional asymmetric random walk whose jumps are
identical, independent and drawn from a distribution \phi(\eta) displaying
asymmetric power law tails (i.e. \phi(\eta) \sim c/\eta^{\alpha +1} for large
positive jumps and \phi(\eta) \sim c/(\gamma |\eta|^{\alpha +1}) for large
negative jumps, with 0 < \alpha < 2). In absence of boundaries and after a
large number of steps n, the probability density function (PDF) of the walker
position, x_n, converges to an asymmetric L\'evy stable law of stability index
\alpha and skewness parameter \beta=(\gamma-1)/(\gamma+1). In particular the
right tail of this PDF decays as c n/x_n^{1+\alpha}. Much less is known when
the walker is confined, or partially confined, in a region of the space. In
this paper we first study the case of a walker constrained to move on the
positive semi-axis and absorbed once it changes sign. In this case, the
persistence exponent \theta_+, which characterizes the algebraic large time
decay of the survival probability, can be computed exactly and we show that the
tail of the PDF of the walker position decays as c \, n/[(1-\theta_+) \,
x_n^{1+\alpha}]. This last result can be generalized in higher dimensions such
as a planar L\'evy walker confined in a wedge with absorbing walls. Our results
are corroborated by precise numerical simulations.
