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- W3042192703 abstract "We study weak convergence of a sequence of point processes to a scale-invariant simple point process. For a deterministic sequence $(z_{n})_{nin mathbb {N}}$ of positive real numbers increasing to infinity as $n to infty $ and a sequence $(X_{k})_{kin mathbb {N}}$ 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 $ct^{-1}$, $tin (0,infty )$. 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." @default.
- W3042192703 created "2020-07-16" @default.
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- W3042192703 date "2020-01-01" @default.
- W3042192703 modified "2023-10-14" @default.
- W3042192703 title "Convergence to scale-invariant Poisson processes and applications in Dickman approximation" @default.
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- W3042192703 doi "https://doi.org/10.1214/20-ejp482" @default.
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