  We study limit laws for simple random walks on supercritical long range
percolation clusters on $\Z^d, d \geq 1$. For the long range percolation model,
the probability that two vertices $x, y$ are connected behaves asymptotically
as $\|x-y\|_2^{-s}$. When $s\in(d, d+1)$, we prove that the scaling limit of
simple random walk on the infinite component converges to an $\alpha$-stable
L\'evy process with $\alpha = s-d$ establishing a conjecture of Berger and
Biskup. The convergence holds in both the quenched and annealed senses. In the
case where $d=1$ and $s>2$ we show that the simple random walk converges to a
Brownian motion. The proof combines heat kernel bounds from our companion
paper, ergodic theory estimates and an involved coupling constructed through
the exploration of a large number of walks on the cluster.
