Invariance principles for changepoint problems (Q1110211)

The asymptotic behavior of certain processes \(U_ k\), obtained from U- statistics, is studied. Motivation for doing this comes from testing the hypothesis \(H_ 0\) that the independent random variables under consideration have the same distribution, versus the alternative \(H_ a\) that at some moment in time \(\lambda\), the sampling distribution has been changed (once through experiment only). Under \(H_ 0\), weak convergence of \(n^{-3/2}U_{[(n+1)t]}/\sigma\) to \((1-t)W_ t+t\{W_ 1-W_ t\},\) where \(\{W_ t\}_{0\leq t\leq 1}\) is a Wiener process, is obtained as \(n\to \infty\). The asymptotic behavior of \(\max_{1\leq k\leq n}U_ k[k(n-k+1)n]^{-1/2}/\sigma\) under \(H_ 0\) and as \(n\to \infty\) is also analyzed.