  We study Brightest Cluster Galaxies (BCGs) in $\sim5000$ galaxy clusters from
the Hyper Suprime-Cam (HSC) Subaru Strategic Program. The sample is selected
over an area of 830 $\textrm{deg}^2$ and is uniformly distributed in redshift
over the range $z=0.3-1.0$. The clusters have stellar masses in the range
$10^{11.8} - 10^{12.9} M_{\odot}$. We compare the stellar mass of the BCGs in
each cluster to what we would expect if their masses were drawn from the mass
distribution of the other member galaxies of the clusters. The BCGs are found
to be "special", in the sense that they are not consistent with being a
statistical extreme of the mass distribution of other cluster galaxies. This
result is robust over the full range of cluster stellar masses and redshifts in
the sample, indicating that BCGs are special up to a redshift of $z=1.0$.
However, BCGs with a large separation from the center of the cluster are found
to be consistent with being statistical extremes of the cluster member mass
distribution. We discuss the implications of these findings for BCG formation
scenarios.
