  Previous studies suggest that the growth of supermassive black holes (SMBHs)
may be fundamentally related to host-galaxy stellar mass ($M_\star$). To
investigate this SMBH growth-$M_\star$ relation in detail, we calculate
long-term SMBH accretion rate as a function of $M_\star$ and redshift
[$\overline{\rm BHAR}(M_\star, z)$] over ranges of
$\log(M_\star/M_\odot)=\text{9.5--12}$ and $z=\text{0.4--4}$. Our
$\overline{\rm BHAR}(M_\star, z)$ is constrained by high-quality survey data
(GOODS-South, GOODS-North, and COSMOS), and by the stellar mass function and
the X-ray luminosity function. At a given $M_\star$, $\overline{\rm BHAR}$ is
higher at high redshift. This redshift dependence is stronger in more massive
systems (for $\log(M_\star/M_\odot)\approx 11.5$, $\overline{\rm BHAR}$ is
three decades higher at $z=4$ than at $z=0.5$), possibly due to AGN feedback.
Our results indicate that the ratio between $\overline{\rm BHAR}$ and average
star formation rate ($\overline{\rm SFR}$) rises toward high $M_\star$ at a
given redshift. This $\overline{\rm BHAR}/\overline{\rm SFR}$ dependence on
$M_\star$ does not support the scenario that SMBH and galaxy growth are in
lockstep. We calculate SMBH mass history [$M_{\rm BH}(z)$] based on our
$\overline{\rm BHAR}(M_\star, z)$ and the $M_\star(z)$ from the literature, and
find that the $M_{\rm BH}$-$M_\star$ relation has weak redshift evolution since
$z\approx 2$. The $M_{\rm BH}/M_\star$ ratio is higher toward massive galaxies:
it rises from $\approx 1/5000$ at $\log M_\star\lesssim 10.5$ to $\approx
1/500$ at $\log M_\star \gtrsim 11.2$. Our predicted $M_{\rm BH}/M_\star$ ratio
at high $M_\star$ is similar to that observed in local giant ellipticals,
suggesting that SMBH growth from mergers is unlikely to dominate over growth
from accretion.
