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

Large deviation asymptotics for a random variable with L\'evy measure supported by $[0, 1]$