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

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.