On the rate of convergence of estimate of the unknown distribution function of the first order autoregression process (Q2736990)

Let observations of the autoregressive series \(U_k=\beta U_{k-1}+\varepsilon_k,\;k=l,\dots,n,\) be given, where \(\beta\) is an unknown nonrandom parameter and \(\varepsilon_k\) are independent, identically distributed random variables with zero mean, finite variance and unknown distribution function \(G(x)\). Let \(\hat G(x)\) be the standard estimate of \(G(x)\). The author proposes an exponential estimate from below for the probability \(P\{\sup_x\sqrt{n}|\hat{G}(x)- G(x)|>\varepsilon\}\).