On convergence of empirical point processes
Let \(\alpha \in (0,\infty)\) and \(\pi_\alpha\) denote a Poisson point process on \(\mathbb{R}^d \setminus \{ 0 \} = S_{d-1} \times (0,\infty)\) with intensity measure \(m_\alpha = \sigma \times \theta_\alpha\), where \(\sigma\) is a finite Borel measure on \(S_{d-1}\) and \(\theta_\alpha\) the measure on \((0,\infty)\) with Lebesgue density \(r \mapsto \alpha r^{-1-\alpha}\). Let \((\xi_n)_{n\geq 1}\) be an i.i.d. sequence of \(\mathbb{R}^d\)-valued random vectors, and assume that \[ \lim_{n\to\infty} n P( \xi_1 / \| \xi_1\| \in A, \, \| \xi_1\| > r b_n) = \sigma(A) \, \theta_\alpha ((r,\infty)) \] for all \(r> 0\) and all Borel subsets \(A\) of \(S_{d-1}\) with \(\sigma\)-boundary zero, where \(b_n\) is of the form \(b_n = n^{1/\alpha} L(n)\) for some slowly varying function \(L\) at infinity. Then it is well known that the scaling empirical process \(\beta_n := \sum_{k=1}^n \delta_{ \xi_k/b_n}\) converges weakly to \(\pi_\alpha\) as \(n\to \infty\). Under the additional assumption that \(\xi_1\) has no atoms, the authors show a stronger type of convergence, namely that \(\beta_n\) converges weakly to \(\pi_\alpha\) with respect to the \(\delta_p\)-metric for \(p > \alpha\). Applications of this result include the study of the asymptotic behaviour of truncated sums of the form \((b_n)^{-1} \sum_{i \leq n, i \in C_n} \xi_i\), where \((C_n)_{n\geq 1}\) is an increasing sequence of subsets of the positive integers.
scientific article; zbMATH DE number 5072670