  We construct `self-stabilizing' processes {Z(t), t $\in [t_0,t_1)$}. These
are random processes which when `localized', that is scaled around t to a fine
limit, have the distribution of an $\alpha$(Z(t))-stable process, where
$\alpha$ is some given function on R. Thus the stability index at t depends on
the value of the process at t. Here we address the case where $\alpha$: R $\to$
(0,1). We first construct deterministic functions which satisfy a kind of
autoregressive property involving sums over a plane point set $\Pi$. Taking
$\Pi$ to be a Poisson point process then defines a random pure jump process,
which we show has the desired localized distributions.
