On the rate of convergence to a Poisson process
Let \(\xi_1, \xi_2,\dots\) be a sequence of i.i.d. spherically symmetric \(\mathbb{R}^d\)-valued random vectors, and consider the associated binomial point process \[ \beta_n = \sum_{j=1}^n \delta_{\frac{\xi_j}{b_n}}. \] This article investigates the rate of convergence of \(\beta_n\) to a Poisson process under the assumption that \(\operatorname{P}(\|\xi_1\| > x)=G(x)\) is continuous, and that, for some \(\alpha >0\), \(\alpha \not = 1\), \(G(x) = x^{-\alpha} M(x)\) and \[ \lim_{x\rightarrow +\infty} M(x) = L^{\alpha} \qquad \text{ for some real number } L > 0, \] or equivalently, that \(G^{-1}(y) = y^{-1/\alpha} L(y)\), with \(L(y)\rightarrow L\) as \(y\rightarrow 0\). Let \(\pi_{\alpha}\) be a Poisson point process on \(S_{d-1}\times (0,+\infty)\) with intensity measure \(m_{\alpha} = \sigma \times \theta_{\alpha}\), where \(\sigma\) is a finite Borel measure on the \((d-1)\)-dimensional unit sphere \(S_{d-1}\), and \(\theta_{\alpha}\) is the measure on \((0,+\infty)\) with Lebesgue density \(r \mapsto \alpha\, r^{-\alpha -1}\). It is known, see [\textit{Y. Davydov} and \textit{V. Egorov}, Stat. Probab. Lett. 76, No. 17, 1836--1844 (2006; Zbl 1124.60045)], that the above assumptions imply that with \(b_n = n^{1/\alpha}\), \(\beta_n \) converges weakly to \(\pi_{\alpha}\) with respect to a metric \(\delta_p\) (of \(l_p\) type in the space of configurations), \(p>\alpha\), which is stronger than vague convergence. Under the additional assumption that \[ \int_0^1 \frac{|1-L(y)/L|^p}{y^{p/\alpha}}\, dy < +\infty, \] the author of this article proves the existence of point processes \(\pi_{\alpha}'\) and \(\beta_n'\), \(n=1,2,\dots\), on a single probability space, equivalent to \(\pi_{\alpha}\) and \(\beta_n\), which almost surely satisfy the estimate \[ \delta_p(\beta_n',\pi_{\alpha}') =\begin{cases} O\left(n^{1/p - 1/\alpha} + n^{-1/2} (\ln \ln n)^{1/2}\right) &\text{ for } \, p>1,\\ O\left(n^{1 -p/\alpha} + n^{-p/2} (\ln \ln n)^{p/2}\right) &\text{ for } \, 0 < p < 1.\end{cases} \] As an application of this result, the author also derives an estimate for the rate of convergence of sums of independent random vectors to the corresponding stable random vector.
scientific article; zbMATH DE number 6109177