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: $X_k>0,\ \ \ \mu_k\equiv EX_k<\infty, \ \ \ \lim_{k\to\infty}\frac{X_k}{\mu_k}\stackrel{\text{dist}}{=}1$. 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 $W_n=\frac1{M_n}\sum_{k=1}^nB_kX_k$. 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 in distribution to a generalized Dickman distribution GD$(\theta)$. In particular, we obtain the following concrete results, which reveal a strange domain of attraction to generalized Dickman distributions. Let $J_\mu,J_p$ be nonnegative integers, let $c_\mu,c_p>0$ and let $$ \begin{aligned} &\mu_n\sim c_\mu n^{a_0}\prod_{j=1}^{J_\mu}(\log^{(j)}n)^{a_j}, &p_n\sim c_p\big({n^{b_0}\prod_{j=1}^{J_p}(\log^{(j)}n)^{b_j}}\big)^{-1}, \ b_{J_p}\neq0. \end{aligned} $$ If $$ \begin{aligned} &i.\ J_p\le J_\mu; &ii.\ b_j=1, \ 0\le j\le J_p; &iii.\ a_j=0, \ 0\le j\le J_p-1,\ \text{and}\ \ a_{J_p}>0, \end{aligned} $$ then $ \lim_{n\to\infty}W_n\stackrel{\text{dist}}{=}\frac1{\theta}\text{GD}(\theta),\ \text{where}\ \theta=\frac{c_p}{a_{J_p}}. $ Otherwise, $\lim_{n\to\infty}W_n\stackrel{\text{dist}}{=}c$, for some $c\in[0,1]$. We also give an application to the statistics of the number of inversions in certain shuffling schemes.