  Using a generalization of the skew-product representation of planar Brownian
motion and the analogue of Spitzer's celebrated asymptotic Theorem for stable
processes due to Bertoin and Werner, for which we provide a new easy proof, we
obtain some limit Theorems for the exit time from a cone of stable processes of
index $\alpha\in(0,2)$. We also study the case $t\rightarrow0$ and we prove
some Laws of the Iterated Logarithm (LIL) for the (well-defined) winding
process associated to our planar stable process.
