  A power-law time-dependent lightcurve for active galactic nuclei (AGNs) is
expected by the self-regulated black hole growth scenario, in which the
feedback of AGNs expels gas and shut down accretion. This is also supported by
the observed power-law Eddington ratio distribution of AGNs. At high redshifts,
the AGN life timescale is comparable with (or even shorter than) the age of the
universe, which set a constraint on the minimal Eddington ratio for AGNs on the
assumption of a power-law AGN lightcurve. The black hole mass function (BHMF)
of AGN relics is calculated by integrating the continuity equation of massive
black hole number density on the assumption of the growth of massive black
holes being dominated by mass accretion with a power-law Eddington ratio
distribution for AGNs. The derived BHMF of AGN relics at z=0 can fit the
measured local mass function of the massive black holes in galaxies quite well,
provided the radiative efficiency ~0.1 and a suitable power-law index for the
Eddington ratio distribution are adopted. In our calculations of the black hole
evolution, the duty cycle of AGN should be less than unity, which requires the
quasar life timescale >0.5 giga-years.
