  This paper explores how the stochastic accretion of planetesimals onto white
dwarfs would be manifested in observations of their atmospheric pollution.
Archival observations of pollution levels for unbiased samples of DA and non-DA
white dwarfs are used to derive the distribution of accretion rates, confirming
that rates become systematically lower as sinking time is decreased, with no
discernable dependence on cooling age. The accretion rates expected from
planetesimals that are all the same mass (ie, a mono-mass distribution) are
explored both analytically and using a Monte Carlo model, quantifying how
measured accretion rates inevitably depend on sinking time, since different
sinking times probe different times since the last accretion event. However,
that dependence is so dramatic that a mono-mass distribution can be excluded.
Consideration of accretion from a broad distribution of planetesimal masses
uncovers an important conceptual difference: accretion is continuous (rather
than stochastic) for planetesimals below a certain mass, and the accretion of
such planetesimals determines the rate typically inferred from observations;
smaller planetesimals dominate the rates for shorter sinking times. A
reasonable fit to the observationally inferred accretion rate distributions is
found with model parameters consistent with a collisionally evolved mass
distribution up to Pluto-mass, and an underlying accretion rate distribution
consistent with that expected from descendants of debris discs of main sequence
A stars. With these parameters, while both DA and non-DA white dwarfs accrete
from the same broad planetesimal distribution, this model predicts that the
pollution seen in DAs is dominated by the continuous accretion of <35km
objects, and that in non-DAs by >35km objects (though the dominant size varies
between stars by around an order of magnitude from this value)...
