This dissertation is concerned with detecting failures in Risk Models and in detecting structural breaks in linear regression models. By applying Theorem 2.1 of Szyszkowicz on U-statistic type process, a number of weak convergence results regarding three weighted partial sum processes are established. It is shown that these partial sum processes share certain invariance properties; estimation risk does not affect their weak convergence results and they are also robust to asymmetries in the error process in linear regression models. There is also an application of the methods developed here to a four factor Capital Asset Pricing model where it is shown via the methods developed in Chapter 3 that manager stock selection abilities vary over time.