Asymptotics for L2-norm of ARCH innovation density estimator

In this paper we consider the asymptotic properties of the ARCH innovation density estimator. We obtain the asymptotic normality of the Bickel–Rosenblatt test statistic (based on our density estimator) under the null hypothesis, which is the same as in the case of the one sample set up (given in Bickel and Rosenblatt, 1973). We also show the strong consistency of the estimator for the true density in L2-norm.