  Limit theorems for the normalized laws with respect to two kinds of weight
functionals are studied for any symmetric stable L\'evy process of index $ 1 <
\alpha \le 2 $. The first kind is a function of the local time at the origin,
and the second kind is the exponential of an occupation time integral. Special
emphasis is put on the role played by a stable L\'evy counterpart of the
universal $ \sigma $-finite measure, found in [9] and [10], which unifies the
corresponding limit theorems in the Brownian setup for which $ \alpha =2 $.
