identifiable subhaloes within it due to tidal stripping. Therefore, we need to account for the mass of the subhaloes not at present, but that which they had at the time of their merger into the parent. The relation between mass and luminosity is then obtained statistically by matching the numbers of galaxies with the total number of hosts, that is, haloes plus subhaloes through their distributions.

The halo abundance is given by the usual Sheth-Tormen mass function [@stmf]:

$$\label{stmf}
n_h(M) dM = A \Big( 1+\frac{1}{\nu^{2q}}\Big) \sqrt{\frac{2}{\pi}} 
\frac{\rho_m}{M} \frac{d\nu}{dM} {\rm exp}\Big(-\frac{\nu^2}{2}\Big) dM\, ,$$

with $\nu=\sqrt{a}\frac{\delta_c}{D(z) \sigma(M)}$, $a=0.707$, $A\approx 0.322$ and $q=0.3$; as usual, $\sigma(M)$ is the variance on the mass scale $M$, $D(z)$ is the growth factor, and $\delta_c$ is the linear threshold for spherical collapse, which in the case of a flat universe is $\delta_c=1.686$, with a small correction dependent on $\Omega_m$ ($\delta_c=1.673$ for $\Omega_m=0.24$).

Following the discussion in paper I, we will assume a very simple model for the subhalo mass distribution within the parent. In terms of their original, pre-accretion mass, we assume that the subhalo distribution is given by a simple Schechter function:

$$\label{shmf}
N(m|M) dm = A(M) (m/\beta M)^{-\alpha} {\rm exp} (-m/\beta M) dm/\beta M \, 