  We give the asymptotics of the tail of the distribution of the first exit
time of the isotropic $\alpha$-stable L\'evy process from the Lipschitz cone in
$\mathbb{R}^d$. We obtain the Yaglom limit for the killed stable process for
the cone. We construct and estimate entrance laws for the process from the
vertex into the cone. For the symmetric Cauchy process and the positive
half-line we give a spectral representation of the Yaglom limit. Our approach
relies on the scalings of the stable process and the cone, which allow to
express the temporal asymptotics of the distribution of the process at infinity
by means of the spatial asymptotics of harmonic functions of the process at the
vertex; on the representation of the probability of survival of the process in
the cone as a Green potential; and on the approximate factorization of the heat
kernel of the cone, which secures compactness and yields a limiting (Yaglom)
measure by means of Prokhorov's theorem.
