  We consider random walks on the mapping class group whose support generates a
non-elementary subgroup and contains a pseudo-Anosov map whose invariant
Teichm\"uller geodesic is in the principal stratum. For such random walks, we
show that mapping classes along almost every infinite sample path are
eventually pseudo-Anosov, with invariant Teichm\"uller geodesics in the
principal stratum. This provides an answer to a question of Kapovich and Pfaff.
