  Asymptotics for Dickman's number theoretic function $\rho(u)$, as $u
\rightarrow \infty$, were given de Bruijn and Alladi, and later in sharper form
by Hildebrand and Tenenbaum. The perspective in these works is that of analytic
number theory. However, the function $\rho(\cdot)$ also arises as a constant
multiple of a certain probability density connected with a scale invariant
Poisson process, and we observe that Dickman asymptotics can be interpreted as
a Gaussian local limit theorem for the sum of arrivals in a tilted Poisson
process, combined with untilting.
  In this paper we exploit and extend this reasoning to obtain analogous
asymptotic formulas for a class of functions including, in addition to
Dickman's function, the densities of random variables having L\'evy measure
with support contained in $[0,1]$, subject to mild regularity assumptions.
