  The fastest tunneling response in double barrier resonant structures is
investigated by considering explicit analytic solutions of the time dependent
Schr\"{o}dinger equation. For cutoff initial plane waves, we find that the
earliest tunneling events consist on the emission of a series of propagating
pulses of the probability density governed by the buildup oscillations in the
quantum well. We show that the fastest tunneling response comes from the
contribution of incident carriers at energies different from resonance, and
that its relevant time scale is given by $\tau_r=\pi \hbar /| E-\epsilon | $,
where $\epsilon $ is the resonance energy and $E$ is the incidence energy.
