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

We study weak convergence of a sequence of point processes to a scale-invariant simple point process. For a deterministic sequence (zn)n∈N of positive real numbers increasing to infinity as n→∞ and a sequence (Xk)k∈N of independent non-negative integer-valued random variables, we consider the sequence of point processes νn=∞∑k=1Xkδzk/zn,n∈N, and prove that, under some general conditions, it converges vaguely in distribution to a scale-invariant Poisson process ηc on (0,∞) with the intensity measure having the density ct−1, t∈(0,∞). An important motivating example from probabilistic number theory relies on choosing Xk∼Geom(1−1/pk) and zk=logpk, k∈N, where (pk)k∈N is an enumeration of the primes in increasing order. We derive a general result on convergence of the integrals ∫10tνn(dt) to the integral ∫10tηc(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,∞) to Rd 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. Citation Download Citation Chinmoy Bhattacharjee. Ilya Molchanov. "Convergence to scale-invariant Poisson processes and applications in Dickman approximation." Electron. J. Probab. 25 1 - 20, 2020. https://doi.org/10.1214/20-EJP482 Information Received: 27 November 2019; Accepted: 8 June 2020; Published: 2020 First available in Project Euclid: 11 July 2020 Zentralblatt MATH identifier: 07252711 Mathematical Reviews number: MR4125784 Digital Object Identifier: 10.1214/20-EJP482 Subjects: Primary: 11K99, 60F05, 60G55, 60G57