On the asymptotic distribution of cramer-von mises one-sample test statistics under an alternative

The asymptotic normality of the Cramer-von Mises one-sample test statistic and one of its variants under an alternative cdf is demonstrated. The derivation herein is unique in that it does not require knowledge of the theory of weak convergence of probability measures defined on metrized function spaces, and thus is accessible to a broader class of students and practitioners.