Bhattacharjee, Chinmoy; Molchanov, Ilya
(2020).
*
Convergence to scale-invariant Poisson processes and applications in Dickman approximation.
*
Electronic journal of probability, 25, pp. 1-20.
Institute of Mathematical Statistics
10.1214/20-EJP482

Text
20-EJP482.pdf - Published Version Restricted to registered users only Available under License Publisher holds Copyright. Download (371kB) | Request a copy |

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

## Item Type: |
Journal Article (Original Article) |
---|---|

## Division/Institute: |
08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science |

## UniBE Contributor: |
Bhattacharjee, Chinmoy and Molchanov, Ilya |

## Subjects: |
300 Social sciences, sociology & anthropology > 360 Social problems & social services 500 Science > 510 Mathematics |

## ISSN: |
1083-6489 |

## Publisher: |
Institute of Mathematical Statistics |

## Language: |
English |

## Submitter: |
Ilya Molchanov |

## Date Deposited: |
25 Feb 2021 15:28 |

## Last Modified: |
26 Feb 2021 02:20 |

## Publisher DOI: |
10.1214/20-EJP482 |

## BORIS DOI: |
10.48350/152418 |

## URI: |
https://boris.unibe.ch/id/eprint/152418 |