Asymptotic results for weighted means of random variables which converge to a Dickman distribution, and some number theoretical applications

¹ Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
giuliano@dm.unipi.it
² Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Rome, Italy
macci@mat.uniroma2.it

Received: 13 January 2014
Revised: 5 August 2014

Abstract

This paper studies some examples of weighted means of random variables. These weighted means generalize the logarithmic means. We consider different kinds of random variables and we prove that they converge weakly to a Dickman distribution; this extends some known results in the literature. In some cases we have interesting connections with number theory. Moreover we prove large deviation principles and, arguing as in [R. Giuliano and C. Macci, J. Math. Anal. Appl. 378 (2011) 555–570], we illustrate how the rate function can be expressed in terms of the Hellinger distance with respect to the (weak) limit, i.e. the Dickman distribution.