  The relation between nuclear ($\lesssim$ 50 pc) star formation and nuclear
galactic activity is still elusive: theoretical models predict a link between
the two, but it is unclear whether active galactic nuclei (AGNs) should appear
at the same time, before or after nuclear star formation activity is ongoing.
We present a study of this relation in a complete, volume-limited sample of
nine of the most luminous ($\log L_{\rm 14-195 keV} > 10^{42.5}$ erg/s) local
AGNs (the LLAMA sample), including a sample of 18 inactive control galaxies (6
star-forming; 12 passive) that are matched by Hubble type, stellar mass (9.5
$\lesssim$ log M_star/M_sun $\lesssim$ 10.5), inclination and distance. This
allows us to calibrate our methods on the control sample and perform a
differential analysis between the AGN and control samples. We perform stellar
population synthesis on VLT/X-SHOOTER spectra in an aperture corresponding to a
physical radius of $\approx$ 150 pc. We find young ($\lesssim$ 30 Myr) stellar
populations in seven out of nine AGNs and in four out of six star-forming
control galaxies. In the non-star-forming control population, in contrast, only
two out of twelve galaxies show such a population. We further show that these
young populations are not indicative of ongoing star-formation, providing
evidence for models that see AGN activity as a consequence of nuclear star
formation. Based on the similar nuclear star-formation histories of AGNs and
star-forming control galaxies, we speculate that the latter may turn into the
former for some fraction of their time. Under this assumption, and making use
of the volume-completeness of our sample, we infer that the AGN phase lasts for
about 5 % of the nuclear starburst phase.
