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: $X_k>0,{\mu }_k\equiv E{X}_k<\infty ,{\lim }_{k\to \infty }\frac{X_k}{{\mu }_k}\stackrel{\text{dist}}{=}1$. Let $M_n={\sum }_{k=1}^n{p}_k{\mu }_k$, assume that $M_n\to \infty$ and define the normalized sum of independent random variables $W_n=\frac{1}{M_n}{\sum }_{k=1}^n{B}_k{X}_k$. 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 $$\begin{array}{rlr}& {\mu }_n\sim c_{\mu }{n}^{a_0}\prod _{j=1}^{J_{\mu }}({\log }^{(j)}n{)}^{a_j},& p_n\sim c_p(n^{b_0}\prod _{j=1}^{J_p}({\log }^{(j)}n{)}^{b_j}{)}^{-1},b_{J_p}\ne 0.\end{array}$$ If $$\begin{array}{rlrl}& i.J_p\le J_{\mu };& ii.b_j=1,0\le j\le J_p;& iii.a_j=0,0\le j\le J_p-1,\text{and}{a}_{J_p}>0,\end{array}$$ 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, ${\lim }_{n\to \infty }{W}_n\stackrel{\text{dist}}{=}c$, for some $c\in [0,1]$. We also give an application to the statistics of the number of inversions in certain shuffling schemes.

arXiv commentThe main result has been expanded a little, and a few minor corrections have been made