Asymptotics of the Lp-norms of density estimators in the first-order autoregressive models

We demonstrate that, for any 1[less-than-or-equals, slant]p<[infinity], the Lp-distance between the kernel density estimators of the residuals and errors in the first-order autoregressive models is so small that the asymptotic behaviour of the Lp-distance between the kernel density estimator of the residuals and the density function itself is the same as in the well-known case concerning the Lp-distance between the kernel density estimator of the i.i.d. errors and the density function.