  We report new results about the anomalous diffusion of a particle in an aging
medium. For each given age, the quasi-stationary particle velocity is governed
by a generalized Langevin equation with a frequency-dependent friction
coefficient proportional to $|\omega|^{\delta-1}$ at small frequencies, with
$0<\delta<2$. The aging properties of the medium are encoded in a frequency
dependent effective temperature $T_{\rm eff.}(\omega)$. The latter is modelized
by a function proportional to $|\omega|^\alpha$ at small frequencies, with
$\alpha<0$, thus allowing for the medium to have a density of slow modes
proportionally larger than in a thermal bath. Using asymptotic Fourier
analysis, we obtain the behaviour at large times of the velocity correlation
function and of the mean square displacement. As a result, the anomalous
diffusion exponent in the aging medium appears to be linked, not only to
$\delta$ as it would be the case in a thermal bath, but also to the exponent
$\alpha$ characterizing the density of slow modes.
