Dickman approximation in simulation, summations and perpetuities

Abstract

The generalized Dickman distribution ${\cal D}_\theta$ with parameter $\theta>0$ is the unique solution to the distributional equality $W=_d W^*$, where \begin{eqnarray} W^*=_d U^{1/\theta}(W+1) \qquad (1) \end{eqnarray} with $W$ non-negative with probability one, $U \sim {\cal U}[0,1]$ independent of $W$, and $=_d$ denoting equality in distribution. Members of this family appear in number theory, stochastic geometry, perpetuities and the study of algorithms. We obtain bounds in Wasserstein type distances between ${\cal D}_\theta$ and \begin{eqnarray} W_n= \frac{1}{n} \sum_{i=1}^n Y_k B_k \qquad (2) \end{eqnarray} where $B_1,\ldots,B_n, Y_1, \ldots, Y_n$ are independent with $B_k \sim {\rm Ber}(1/k), E[Y_k]=k, {\rm Var}(Y_k)=\sigma_k^2$ and provide an application to the minimal directed spanning tree in $\mathbb{R}^2$, and also obtain such bounds when the Bernoulli variables in $(2)$ are replaced by Poissons. We also give simple proofs and provide bounds with optimal rates for the Dickman convergence of the weighted sums, arising in probabilistic number theory, of the form \begin{eqnarray} S_n=\frac{1}{\log(p_n)} \sum_{k=1}^n X_k \log(p_k) \end{eqnarray} where $(p_k)_{k \ge 1}$ is an enumeration of the prime numbers in increasing order and $X_k$ is Geometric with parameter $(1-1/p_k)$, Bernoulli with success probability $1/(1+p_k)$ or Poisson with mean $\lambda_k$. In addition, we broaden the class of generalized Dickman distributions by studying the fixed points of the transformation \begin{eqnarray*} s(W^*)=_d U^{1/\theta}s(W+1) \end{eqnarray*} generalizing $(1)$, that allows the use of non-identity utility functions $s(\cdot)$ in Vervaat perpetuities. We obtain distributional bounds for recursive methods that can be used to simulate from this family.