  We consider random walk $(X_n)_{n\geq0}$ on $\mathbb{Z}^d$ in a space--time
product environment $\omega\in\Omega$. We take the point of view of the
particle and focus on the environment Markov chain $(T_{n,X_n}\omega)_{n\geq0}$
where $T$ denotes the shift on $\Omega$. Conditioned on the particle having
asymptotic mean velocity equal to any given $\xi$, we show that the empirical
process of the environment Markov chain converges to a stationary process
$\mu_{\xi}^{\infty}$ under the averaged measure. When $d\geq3$ and $\xi$ is
sufficiently close to the typical velocity, we prove that averaged and quenched
large deviations are equivalent and when conditioned on the particle having
asymptotic mean velocity $\xi$, the empirical process of the environment Markov
chain converges to $\mu_{\xi}^{\infty}$ under the quenched measure as well. In
this case, we show that $\mu_{\xi}^{\infty}$ is a stationary Markov process
whose kernel is obtained from the original kernel by a Doob $h$-transform.
