  We study the aging behavior of a truncated version of the Random Energy Model
evolving under Metropolis dynamics. We prove that the natural time-time
correlation function defined through the overlap function converges to an
arcsine law distribution function, almost surely in the random environment and
in the full range of time scales and temperatures for which such a result can
be expected to hold. This establishes that the dynamics ages in the same way as
Bouchaud's REM-like trap model, thus extending the universality class of the
latter model. The proof relies on a clock process convergence result of a new
type where the number of summands is itself a clock process. This reflects the
fact that the exploration process of Metropolis dynamics is itself an aging
process, governed by its own clock. Both clock processes are shown to converge
to stable subordinators below certain critical lines in their time-scale and
temperature domains, almost surely in the random environment.
