  The physics of primordial black holes can be affected by the non-Gaussian
statistics of the density fluctuations that generate them. Therefore, it is
important to have good theoretical control of the higher-order correlation
functions for primordial curvature perturbations. By working at leading order
in a $1/|\eta|$ expansion, we analytically determine the bispectrum of
curvature fluctuations for single field inflationary scenarios producing
primordial black holes. The bispectrum has a rich scale and shape dependence,
and its features depend on the dynamics of the would-be decaying mode. We apply
our analytical results to study gravitational waves induced at second order by
enhanced curvature fluctuations. Their statistical properties are derived in
terms of convolution integrals over wide momentum ranges, and they are
sensitive on the scale and shape dependence of the curvature bispectrum we
analytically computed.
