Electronic Journal of Probability

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

Ross G Pinsky

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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

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

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

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems

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

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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

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