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

We consider parameter
estimation for a class of ARCH(infinity) 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(infinity) parameters tend to zero.

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