ASYMPTOTIC RESULTS FOR WEIGHTED MEANS OF RANDOM VARIABLES WHICH CONVERGE TO A DICKMAN DISTRIBUTION, AND SOME NUMBER THEORETICAL APPLICATIONS

被引：7

作者：

Giuliano, Rita ^{[1
]}

Macci, Claudio ^{[2
]}

机构：

[1] Univ Pisa, Dipartimento Matemat, I-56127 Pisa, Italy

[2] Univ Roma Tor Vergata, Dipartimento Matemat, I-00133 Rome, Italy

来源：

ESAIM-PROBABILITY AND STATISTICS | 2015年 / 19卷

关键词：

Almost sure central limit theorem; Dickman function; Hellinger distance; large deviations; prime numbers; square-free numbers; LARGE DEVIATION PRINCIPLE; SURE CONVERGENCE; LIMIT; SUMS;

D O I：

10.1051/ps/2014030

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

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.

引用

页码：395 / 413

页数：19

共 37 条

[1]

[Anonymous], PATHWISE CENTRAL LIM

[2] INDEPENDENT PROCESS APPROXIMATIONS FOR RANDOM COMBINATORIAL STRUCTURES [J].

ARRATIA, R ;

TAVARE, S .

ADVANCES IN MATHEMATICS, 1994, 104 (01) :90-154

[3]

Atlagh M., 2000, EXPO MATH, V18, P097

[4] A universal result in almost sure central limit theory [J].

Berkes, I ;

Csáki, E .

STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 2001, 94 (01) :105-134

[5] AN ALMOST EVERYWHERE CENTRAL LIMIT-THEOREM [J].

BROSAMLER, GA .

MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 1988, 104 :561-574

[6] Non-standard limit theorems in number theory [J].

Cellarosi, F. (fcellaro@math.princeton.edu), 1600, Springer Science and Business Media, LLC (33) :197-213

[7] Almost sure convergence in extreme value theory [J].

Cheng, SH ;

Peng, L ;

Qi, YC .

MATHEMATISCHE NACHRICHTEN, 1998, 190 :43-50

[8]

De Koninck JM, 2012, J INTEGER SEQ, V15

[9]

Dembo A., 2009, Large Deviations Techniques and Applications, V2nd

[10]

Dickman K., 1930, ARK MAT ASTRON FYS, V22, P1

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