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

TL;DR

This paper investigates the conditions under which normalized sums of independent Bernoulli-weighted positive random variables converge to either a constant or a generalized Dickman distribution, revealing a peculiar domain of attraction.

Contribution

It establishes general criteria for convergence to generalized Dickman distributions in sums of independent variables, including explicit asymptotic forms and applications to permutation statistics.

Findings

Identifies conditions for convergence to GD(θ) or a constant.

Provides explicit asymptotic forms involving iterated logarithms.

Applies results to inversion counts in shuffling schemes.

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.