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

Let {B_k}^∞_k=1,{X_k}^∞_k=1 all be independent random variables. Assume that {B_k}^∞_k=1 are {0,1}-valued Bernoulli random variables satisfying B_k^distBer(p_k), with ∑^∞_k=1p_k=∞, and assume that {X_k}^∞_k=1 satisfy: X_k>0, μ_k≡EX_k<∞, limk→∞Xkμk=dist1. Let M_n=∑ⁿ_k=1p_kμ_k, assume that Mn→∞ and define the normalized sum of independent random variables W_n=1/Mn∑ⁿ_k=1B_kX_k. We give a general condition under which W_n^→distc, for some c∈[0,1], and a general condition under which W_n converges in distribution to a generalized Dickman distribution GD(θ). In particular, we obtain the following concrete results, which reveal a strange domain of attraction to generalized Dickman distributions. Let J_μ, J_p be nonnegative integers, let c_μ,c_p>0 and let (Formula presented) If (Formula presented) then. Otherwise, (Formula presented) depends on the above parameters. We also give an application to the statistics of the number of inversions in certain shuffling schemes.