Electronic Journal of Probability

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

Ross G Pinsky

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

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

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

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