  Let $(X_t, t \geq 0)$ be an $\alpha$-stable random walk with values in
$\Z^d$. Let $l_t(x) = \int_0^t \delta_x(X_s) ds$ be its local time. For $p>1$,
not necessarily integer, $I_t = \sum_x l_t^p(x)$ is the so-called $p$-fold
self- intersection local time of the random walk. When $p(d -\alpha) < d$, we
derive precise logarithmic asymptotics of the probability $P(I_t \geq r_t)$ for
all scales $r_t \gg \E(I_t)$. Our result extends previous works by Chen, Li and
Rosen 2005, Becker and K\"onig 2010, and Laurent 2012.
