Convergence to scale-invariant Poisson processes and applications in Dickman approximation. (English) Zbl 1459.60106

Bhattacharjee, Chinmoy; Molchanov, Ilya

Electron. J. Probab. 25, Paper No. 79, 20 p. (2020).

Summary: 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 \(X_k \sim{\text{Geom}}(1-1/p_k)\) 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.

MSC:

60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05	Central limit and other weak theorems
60G57	Random measures

Keywords:

Poisson processes; vague convergence; scale invariance; random measures; Dickman distributions

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