The stratum of random mapping classes

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.