Dickman approximation in simulation, summations and perpetuities

Abstract

The generalized Dickman distribution $\mathcal{D}_{\theta}$ with parameter $\theta>0$ is the unique solution to the distributional equality $W=_{d}W^{*}$, where \begin{align} W^∗=_d U^1/θ(W+1),\tag{1} \end{align} with $W$ non-negative with probability one, $U\sim\mathcal{U}[0,1]$ independent of $W$, and $=_{d}$ denoting equality in distribution. These distributions appear in number theory, stochastic geometry, perpetuities and the study of algorithms. We obtain bounds in Wasserstein type distances between $\mathcal{D}_{\theta}$ and the distribution of \begin{eqnarray*}W_{n}=\frac{1}{n}\sum_{i=1}^{n}Y_{k}B_{k},\end{eqnarray*} where $B_{1},\ldots,B_{n},Y_{1},\ldots,Y_{n}$ are independent with $B_{k}$ distributed $\operatorname{Ber}(1/k)$ or $\mathcal{P}(\theta/k)$, $E[Y_{k}]=k$ and $\operatorname{Var}(Y_{k})=\sigma_{k}^{2}$, and provide an application to the minimal directed spanning tree in $\mathbb{R}^{2}$. We also provide bounds with optimal rates for the Dickman convergence of 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 $(p_{k})_{k\ge1}$ 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 $\lambda_{k}$.

Lastly, 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.