Large deviation asymptotics for a random variable with Lévy measure supported by $[0, 1]$

Abstract

Asymptotics for Dickman's number theoretic function ρ(u), as u → ∞, 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 ρ(•) 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 Lévy measure with support contained in [0, 1], subject to mild regularity assumptions.

On sequential selection and a first passage problem for the Poisson process

“…and that the difference is always less than 1. This contrasts sharply with the second terms in (1) and (2). For the difference between the prophet and gambler values we have therefore EN(t) − EL(t) ∼ 1 12 log t.…”

“…To keep the discussion short, details of routine proofs are only sketched. Related work on sums of consequitive arrivals in the case of inhomogeneous rate appeared in [2], and on the integrated Poisson process in [22].…”