A note on uniform convergence of an ARCH(∞) estimator

We consider parameter estimation for a class of ARCH(∞) models, which do not necessarily have a parametric form. The estimation is based on a normalised least squares approach, where the normalisation is the weighted sum of past observations. The number of parameters estimated depends on the sample size and increases as the sample size grows. Using maximal inequalities for martingales and mixingales we derive a uniform rate of convergence for the parameter estimator. We show that the rate of convergence depends both on the number of parameters estimated and the rate that the ARCH(∞) parameters tend to zero.