  We study a $d$-dimensional non-symmetric strictly $\alpha$-stable L\'{e}vy
process $\mathbf{X}$, whose spherical density is bounded and bounded away from
the origin. First, we give sharp two-sided estimates on the transition density
of $\mathbf{X}$ killed when leaving an arbitrary $\kappa$-fat set. We apply
these results to get the existence of the Yaglom limit for arbitrary
$\kappa$-fat cone. In the meantime we also obtain the spacial asymptotics of
the survival probability at the vertex of the cone expressed by means of the
Martin kernel for $\Gamma$ and its homogeneity exponent. Our results hold for
the dual process $\widehat{\mathbf{X}}$, too.
