  In this paper we study a random walk in a one-dimensional dynamic random
environment consisting of a collection of independent particles performing
simple symmetric random walks in a Poisson equilibrium with density $\rho \in
(0,\infty)$. At each step the random walk performs a nearest-neighbour jump,
moving to the right with probability $p_{\circ}$ when it is on a vacant site
and probability $p_{\bullet}$ when it is on an occupied site. Assuming that
$p_\circ \in (0,1)$ and $p_\bullet \neq \tfrac12$, we show that the position of
the random walk satisfies a strong law of large numbers, a functional central
limit theorem and a large deviation bound, provided $\rho$ is large enough. The
proof is based on the construction of a renewal structure together with a
multiscale renormalisation argument.
