Electronic Journal of Probability

On the strange domain of attraction to generalized Dickman distributions for sums of independent random variables

Ross G Pinsky

Full-text: Open access

Abstract

Let $\{B_k\}_{k=1}^\infty , \{X_k\}_{k=1}^\infty $ all be independent random variables. Assume that $\{B_k\}_{k=1}^\infty $ are $\{0,1\}$-valued Bernoulli random variables satisfying $B_k\stackrel{\text {dist}} {=}\text{Ber} (p_k)$, with $\sum _{k=1}^\infty p_k=\infty $, and assume that $\{X_k\}_{k=1}^\infty $ satisfy $X_k>0$ and $\mu _k\equiv EX_k<\infty $. Let $M_n=\sum _{k=1}^np_k\mu _k$, assume that $M_n\to \infty $ and define the normalized sum of independent random variables $W_n=\frac 1{M_n}\sum _{k=1}^nB_kX_k$. We give a general condition under which $W_n\stackrel{\text {dist}} {\to }c$, for some $c\in [0,1]$, and a general condition under which $W_n$ converges weakly to a distribution from a family of distributions that includes the generalized Dickman distributions GD$(\theta ),\theta >0$. In particular, we obtain the following result, which reveals a strange domain of attraction to generalized Dickman distributions. Assume that $\lim _{k\to \infty }\frac{X_k} {\mu _k}\stackrel{\text {dist}} {=}1$. Let $J_\mu ,J_p$ be nonnegative integers, let $c_\mu ,c_p>0$ and let

$ \mu _n\sim c_\mu n^{a_0}\prod _{j=1}^{J_\mu }(\log ^{(j)}n)^{a_j},\ p_n\sim c_p\big ({n^{b_0}\prod _{j=1}^{J_p}(\log ^{(j)}n)^{b_j}}\big )^{-1}, \ b_{J_p}\neq 0, $ where $\log ^{(j)}$ denotes the $j$th iterate of the logarithm.

If \[ i.\ J_p\le J_\mu ;\\ ii.\ b_j=1, \ 0\le j\le J_p;\\ iii.\ a_j=0, \ 0\le j\le J_p-1,\ \text{and} \ \ a_{J_p}>0, \] then $\lim _{n\to \infty }W_n\stackrel{\text {dist}} {=}\frac 1{\theta }\text{GD} (\theta ),\ \text{where} \ \theta =\frac{c_p} {a_{J_p}}. $

Otherwise, $\lim _{n\to \infty }W_n\stackrel{\text {dist}} {=}\delta _c$, where $c\in \{0,1\}$ depends on the above parameters.

We also give an application to the statistics of the number of inversions in certain random shuffling schemes.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 3, 17 pp.

Dates
Received: 30 January 2017
Accepted: 14 November 2017
First available in Project Euclid: 16 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1516093311

Digital Object Identifier
doi:10.1214/17-EJP126

Mathematical Reviews number (MathSciNet)
MR3751078

Zentralblatt MATH identifier
1390.60092

Subjects
Primary: 60F05: Central limit and other weak theorems

Keywords
Dickman function generalized Dickman distribution domain of attraction normalized sums of independent random variables

Rights
Creative Commons Attribution 4.0 International License.

Citation

Pinsky, Ross G. On the strange domain of attraction to generalized Dickman distributions for sums of independent random variables. Electron. J. Probab. 23 (2018), paper no. 3, 17 pp. doi:10.1214/17-EJP126. https://projecteuclid.org/euclid.ejp/1516093311


Export citation

References

  • [1] Arratia, R., Barbour, A. and Tavaré, S., Logarithmic Combinatorial Structures: A Probabilistic Approach, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zurich, (2003).
  • [2] de Bruijn, N., On the number of positive integers $\leq x$ and free of prime factors $>y$, Nederl. Acad. Wetensch. Proc. Ser. A. 54, (1951), 50–60.
  • [3] Cellarosi, F. and Sinai, Y., Non-standard limit theorems in number theory, Prokhorov and Contemporary Probability Theory, 197–213, Springer Proc. Math. Stat., 33, Springer, Heidelberg, (2013).
  • [4] Dickman, K., On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Math. Astr. Fys. 22, 1–14 (1930).
  • [5] Hwang, H.-K. and Tsai, T.-H. Quickselect and the Dickman function, Combin. Probab. Comput. 11 (2002), 353–371.
  • [6] Montgomery, H. and Vaughan, R., Multiplicative Number Theory. I. Classical Theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, Cambridge, (2007).
  • [7] Penrose, M. and Wade, A., Random minimal directed spanning trees and Dickman-type distributions, Adv. in Appl. Probab. 36 (2004), 691–714.
  • [8] Pinsky R., A Natural Probabilistic Model on the Integers and its Relation to Dickman-Type Distributions and Buchstab’s Function, preprint, (2016). arXiv:1606.02965
  • [9] Tenenbaum, G., Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, 46, Cambridge University Press, Cambridge, (1995). MR1342300