  For a class of additive processes driven by the affine recursion $X_{n+1} =
A_n X_n + B_n$, we develop a sample-path large deviations principle in the
$M_1'$ topology on $D [0,1]$. We allow $B_n$ to have both signs and focus on
the case where Kesten's condition holds on $A_1$, leading to heavy-tailed
distributions. The most likely paths in our large deviations results are step
functions with both positive and negative jumps.
