  Let $(X_t,t\geq 0)$ be a random walk on $\mathbb{Z}^d$. Let $ l_t(x)=
\int_0^t \delta_x(X_s)ds$ be the local time at site $x$ and $ I_t=
\sum\limits_{x\in\mathbb{Z}^d} l_t(x)^p $ the p-fold self-intersection local
time (SILT). Becker and K\"onig have recently proved a large deviations
principle for $I_t$ for all $(p,d)\in\mathbb{R}^d\times\mathbb{Z}^d$ such that
$p(d-2)<2$. We extend these results to a broader scale of deviations and to the
whole subcritical domain $p(d-2)<d$. Moreover we unify the proofs of the large
deviations principle using a method introduced by Castell for the critical case
$p(d-2)=d$ and developed by Laurent for the critical and supercritical case
$p(d-\alpha)\geq d$ of $\alpha$-stable random walk.
