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.
Categories: math.PR