  Polluted white dwarfs are generally accreting terrestrial-like material that
may originate from a debris belt like the asteroid belt in the solar system.
The fraction of white dwarfs that are polluted drops off significantly for
white dwarfs with masses $M_{\rm WD}\gtrsim 0.8\,\rm M_\odot$. This implies
that asteroid belts and planetary systems around main-sequence stars with mass
$M_{\rm MS}\gtrsim 3\,\rm M_\odot$ may not form because of the intense
radiation from the star. This is in agreement with current debris disc and
exoplanet observations. The fraction of white dwarfs that show pollution also
drops off significantly for low mass white dwarfs $(M_{\rm WD}\lesssim
0.55\,\rm M_\odot)$. However, the low-mass white dwarfs that do show pollution
are not currently accreting but have accreted in the past. We suggest that
asteroid belts around main sequence stars with masses $M_{\rm MS}\lesssim
2\,\rm M_\odot$ are not likely to survive the stellar evolution process. The
destruction likely occurs during the AGB phase and could be the result of
interactions of the asteroids with the stellar wind, the high radiation or, for
the lowest mass stars that have an unusually close-in asteroid belt, scattering
during the tidal orbital decay of the inner planetary system.
