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

Let $\{B_k{\}}_{k=1}^{\mathrm{\infty }},\{X_k{\}}_{k=1}^{\mathrm{\infty }}$ all be independent random variables. Assume that $\{B_k{\}}_{k=1}^{\mathrm{\infty }}$ are $\{0,1\}$-valued Bernoulli random variables satisfying $B_k\text{Ber}(p_k)$, with ${\sum }_{k=1}^{\mathrm{\infty }}{p}_k=\mathrm{\infty }$, and assume that $\{X_k{\}}_{k=1}^{\mathrm{\infty }}$ satisfy: $X_k>0,\ \ \ \mu_k\equiv EX_k<\infty, \ \
\lim_{k\to\infty}\frac{X_k}{\mu_k}\stackrel{$\text{dist}$}{=}1$\mathrm{.}Let$M_n=\sum_{k=1}^np_k\mu_k$,assumethat$M_n\to\infty$ and define the normalized sum of independent random variables $W_n=\frac{1}{M_n}{\sum }_{k=1}^n{B}_k{X}_k$. We give a general condition under which $W_{n}c$, 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 $$\begin{array}{rlr}{\displaystyle }& {\displaystyle {\mu }_n\sim c_{\mu }{n}^{a_0}\prod _{j=1}^{J_{\mu }}({\log }^{(j)}n{)}^{a_j},}& {\displaystyle p_n\sim c_p(n^{b_0}\prod _{j=1}^{J_p}({\log }^{(j)}n{)}^{b_j}{)}^{-1},b_{J_p}\mathrm{\ne }0.}\end{array}$$ If $$\begin{array}{rlrl}{\displaystyle }& {\displaystyle i\mathrm{.}{J}_p\le J_{\mu };}& {\displaystyle ii\mathrm{.}{b}_j=1,0\le j\le J_p;}& {\displaystyle iii\mathrm{.}{a}_j=0,0\le j\le J_p-1,\text{ and }{a}_{J_p}>0,}\end{array}$$ 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$,forsome$c\in[0,1]$. We also give an application to the statistics of the number of inversions in certain shuffling schemes.