  A novel statistic is proposed to examine the hypothesis that all cluster
galaxies are drawn from the same luminosity distribution (LD). In such a
"statistical model" of galaxy LD, the brightest cluster galaxies (BCGs) are
simply the statistical extreme of the galaxy population. Using a large sample
of nearby clusters, we show that BCGs in high luminosity clusters (e.g., L_tot
> 4x10^11 L_sun) are unlikely (probability <3x10^-4) to be drawn from the LD
defined by all red cluster galaxies more luminous than M_r=-20. On the other
hand, BCGs in less luminous clusters are consistent with being the statistical
extreme. Applying our method to the second brightest galaxies, we show that
they are consistent with being the statistical extreme, which implies that the
BCGs are also distinct from non-BCG luminous, red, cluster galaxies. We point
out some issues with the interpretation of the classical tests proposed by
Tremaine & Richstone (1977) that are designed to examine the statistical nature
of BCGs, investigate the robustness of both our statistical test and those of
TR against difficulties in photometry of galaxies of large angular size, and
discuss the implication of our findings on surveys that use the luminous red
galaxies to measure the baryon acoustic oscillation features in the galaxy
power spectrum.
