An Extension of the Csörgő-Horváth Functional Limit Theorem and Its Applications to Changepoint Problems

Consider a triangular array Xn1, ..., Xnn, n∈ N, of rowwise independent random elements with values in a measurable space. Suppose there exists θ ∈ [0, 1)such that Xn1, ..., Xn[nθ ] have distribution ν1 and Xn[nθ] + 1(n), ..., Xnn have distribution ν2. Csörgő and Horváth derived an invariance principle for a one-time parameter process, which is the foundation of a test for H0: θ = 0 versus H1: θ ∈(0, 1). We are interested in the more complex test problem H̃0: θ ∈ Θ0 versus H̃1,: θ ∈ Θ0, where Θ0⊆(0, 1). To treat this new situation, we extend the Csörgő-Horváth result in proving a functional limit theorem for a suitable two-time parameter process. We briefly sketch several applications of our result. Especially, the power of the Csörgő-Horváth test is investigated in detail.