An extension of the Csörgő-Horváth functional limit theorem and its applications to changepoint problems
Let \(X^ n_ 1, \dots, X^ n_ n\), \(n \geq 1\), be a triangular array of rowwise independent random variables with values in a measurable space. Suppose there exists \(\theta \in[0,1)\) such that \(X^ n_ 1, \dots, X^ n_{[n \theta]}\) and \(X^ n_{[n \theta] + 1}, \dots, X^ n_ n\) have different unknown distributions. Let \(\Theta \subseteq (0,1]\). To test the hypothesis \(H_ 0 : \theta \in \Theta\) versus \(H_ 1 : \theta \notin \Theta\) the author proves a functional limit theorem for a suitable two-time parameter process \(\{b_ n : n \geq 1\}\). The proof is based on establishing tightness and convergence of the finite- dimensional distribution of \(\{b_ n : n \geq 1\}\) in the space \(D[0,1]^ 2\). Several applications to changepoint problems are given.
scientific article; zbMATH DE number 701321