  QSOs are believed to be powered by accretion onto massive black holes (BHs).
In this paper, assuming that each central BH in nearby galaxies has experienced
the QSO phase and ignoring BH mergers, we establish a relation between the QSO
luminosity function (LF) and the local BH mass function (MF). The QSOLF is
jointly controlled by the luminosity evolution of individual QSOs and the
triggering history of the accretion onto seed BHs. By comparing the time
integral of the QSOLF with that inferred from local BHs, we separate the effect
of the luminosity evolution of individual QSOs from the effect of the
triggering history. Assuming that the nuclear luminosity evolution includes two
phases (first increasing at the Eddington luminosity with growth of BHs and
then declining), we find that observations are generally consistent with the
expected relation between the QSOLF and the local BHMF and obtain the
constraints on QSO models and BH growth. We point out that it is hard to
accurately estimate the value of the QSO lifetime from the QSOLF and/or the
local BHMF, if it is longer than a certain value (e.g., four times of the
Salpeter timescale in this study). We discuss the importance of accurate
measurements of the intrinsic scatter in the BH mass and velocity dispersion
relation of local galaxies and the scatter in the bolometric correction of
QSOs. We also discuss some possible applications of the work in this paper,
such as to the study of the demography of QSOs and the demography of normal
galaxies at intermediate redshift (abridged).

Astrophysics