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

Giuliano, Rita; Macci, Claudio

ESAIM: Probability and Statistics, Tome 19 (2015), pp. 395-413.

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.

Classification : 60F10, 60F05, 11K99
Mots clés : Almost sure central limit theorem, Dickman function, Hellinger distance, large deviations, prime numbers, square-free numbers

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