Strong Consistency of the Kernel Estimator of Error CDF in the Autoregressive Model

We consider the strong convergence rate of the kernel estimation of the cumulative distribution function(CDF). First, the law of the iterated logarithm (LIL) for Lp-norms ofempirical processes is extended to the kernel estimator of the CDF.Then we develop strong convergence of the kernelestimator of the error CDF in the first-order autoregressive model. The LIL for Lp-norm isextended to the residual-based kernel error CDF estimator.It shows that the rate of convergence of a kernel smoothedversion of the residual-based empirical distribution function matches exactly the rates obtained for an independentsample from the error distribution.