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 \[ \nu _{n}=\sum _{k=1}^{\infty }X_{k} \delta _{z_{k}/z_{n}}, \quad n \in \mathbb {N}, \] 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 $ct^{-1}$, $t\in (0,\infty )$. 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.