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

TL;DR

This paper extends asymptotic analysis of Dickman's function to a broader class of densities linked to Lévy measures supported on [0,1], using probabilistic and local limit theorem techniques.

Contribution

It introduces a new probabilistic approach to derive large deviation asymptotics for densities associated with Lévy measures supported on [0,1], generalizing previous number-theoretic results.

Findings

Derived asymptotic formulas for densities with Lévy measures on [0,1]

Connected number theory functions with Poisson process asymptotics

Extended local limit theorem techniques to new class of densities

Abstract

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.