  We study the object formally defined as
\gamma\big([0,t]^{2}\big)=\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-\sigma} dr
ds-E\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-\sigma} dr ds, where $X_{t}$ is the
symmetric stable processes of index $0<\beta\le 2$ in $R^{d}$. When
$\beta\le\sigma<\displaystyle\min \Big\{{3\over 2}\beta, d\Big\}$, this has to
be defined as a limit, in the spirit of renormalized self-intersection local
time. We obtain results about the large deviations and laws of the iterated
logarithm for $\gamma$. This is applied to obtain results about stable
processes in random potentials.
