Convergence to scale-invariant Poisson processes and applications in Dickman approximation

Abstract

We study weak convergence of a sequence of point processes to a scale-invariant simple point process. For a deterministic sequence $(z_n{)}_{n\in \mathbb{N}}$ of positive real numbers increasing to infinity as $n\to \infty$ and a sequence $(X_k{)}_{k\in \mathbb{N}}$ of independent non-negative integer-valued random variables, we consider the sequence of point processes \begin{equation*} ν_n=\sum_{k=1}^\infty X_k δ_{z_k/z_n}, \quad n\in \mathbb{N}, \end{equation*} and prove that, under some general conditions, it converges vaguely in distribution to a scale-invariant Poisson process ${\eta }_c$ on $(0,\infty )$ with the intensity measure having the density $c{t}^{-1}$, $t\in (0,\infty )$. An important motivating example from probabilistic number theory relies on choosing $X_k\sim \mathrm{Geom}(1-1/p_k)$ and $z_k=\log p_k$, $k\in \mathbb{N}$, where $(p_k{)}_{k\in \mathbb{N}}$ is an enumeration of the primes in increasing order. We derive a general result on convergence of the integrals ${\int }_0^{1}t{\nu }_n(dt)$ to the integral ${\int }_0^{1}t{\eta }_c(dt)$, the latter having a generalized Dickman distribution, thus providing a new way of proving Dickman convergence results. We extend our results to the multivariate setting and provide sufficient conditions for vague convergence in distribution for a broad class of sequences of point processes obtained by mapping the points from $(0,\infty )$ to ${\mathbb{R}}^d$ via multiplication by i.i.d. random vectors. In addition, we introduce a new class of multivariate Dickman distributions which naturally extends the univariate setting.

Final version, to appear in Electronic Journal of Probability