  We study partially hyperbolic sets $K$ on a Riemannian manifold $M$ whose
tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the
center-unstable direction $E^{cu}$ is non-uniformly expanding on some local
unstable disk. We prove that the (stretched) exponential decay of recurrence
times for an induced scheme can be deduced under the assumption of (stretched)
exponential decay of the time that typical points need to achieve some uniform
expanding in the center-unstable direction. This extends a result by Alves and
Pinheiro to the (stretched) exponential case. As an application of our main
result we obtain (stretched) exponential decay of correlations and
exponentially large deviations for a class of partially hyperbolic
diffeomorphisms introduced by Alves, Bonatti and Viana.
