A note on Poisson approximation of rescaled set-indexed empirical processes
The author studies Poisson approximation to the multivariate empirical process in a small neighbourhood of 0. Given i.i.d. random variables \(\{X_i,i\geq 1\}\) with values in \(\mathbb{R}^k\) define the point process \(S_n(A)= \sum_{i\leq n}I (n^{1/k} X_i\in A)\), \(A\in{\mathcal B}_0\), where \({\mathcal B}_0\) is the class of all Borel subsets of some bounded set \(A_0\subset \mathbb{R}^k\). If \(X_i\) has a density \(f\) which is continuous at 0, the process \(S_n\) can be defined on the same probability space as a Poisson process \(\Pi\) with intensity \(f(0)\cdot \lambda\) in such a way that \(P(\sup_{A\in{\mathcal B}_0}|S_n(A)-\Pi (A)|> 0)\to 0.\) The proof uses Le Cam's result on Poisson approximation in separable normed spaces to get a bound on the total variation distance between \(S_n\) and \(\Pi\) and Dohrushin's coupling lemma.
scientific article; zbMATH DE number 1395086