On exact rates of convergence in functional limit theorems for \(U\)- statistic type processes
Let \(X,X_ 1, X_ 2, \dots\) be an i.i.d. sequence of random variables taking values in a measurable space. Let \(K(x,y)\) be a symmetric (either antisymmetric) measurable function. Consider the partial sums \(\sum^ n_{i=k+1} \sum^ k_{j=1} K(X_ i,X_ j)\), for integers \(0 \leq k \leq n\), related to the \(U\)-statistics kernel function \(K\). Similarly to the invariance principle, continuous random polygonal lines may be defined. Under certain conditions the random \(U\)-statistic type process with the polygonal lines as sample functions converges in distribution as \(n \to \infty\) to a Gauassian process related to the Brownian motion. The author obtains Berry-Esseen bounds for the convergence rate of these processes in the Prokhorov distance. The optimality of bounds is shown. Applications for the behavior of changepoint estimators are given. Actually the author considers the more general situation of triangular rowwise i.i.d. arrays \(X_{1n}, \dots, X_{nn}\). The main idea of the proof is a reduction to certain partial sum processes.
scientific article; zbMATH DE number 687642