  We use the "Millennium Simulation" to study the mass function of accreted
sub-halos during merger events in the dark halo assembly history. Our study
includes three kinds of sub-halo mergers: (1) mergers that happen to the main
progenitor of dark halos; (2) mergers that happen on the entire merging history
tree of dark halos; and (3) mergers that leave identifiable sub-halos in
present-day dark halos. We estimate the unevolved sub-halo mass functions
(USMFs), for which sub-halo masses are measured at the times of their
accretion. For sub-halos that merge into the main branch of a present-day dark
halo, their USMF can be well described by a universal functional form, in
excellent agreement with previous results. The same conclusion can also be
reached for the USMF of all progenitors that have merged to become sub-halos
during the entire halo merging history. In both cases, the USMFs are also
independent of the redshift of host halos. Due to tidal disruption, only a
small fraction of the accreted halos survive as sub-halos identifiable in the
present-day dark halos. In cluster-sized halos, about 30% of the survived
sub-halos are sub-subhalos, and this fraction decreases with decreasing halo
mass. For given halo and sub-halo masses, the accretion time has very broad
distribution, but the survived sub-halos are all accreted quite recently.
