A sequence of random vectors elements of which depend on time-delayed observations of an autoregressive process is considered and the distribution of smooth functions of the sample mean of such vectors is studied asymptotically. Both classical approximation based on the Edgeworth expansion and the bootstrap distribution are developed. It is shown that the accuracy of bootstrap approximation is $ o(n^{-\frac{1}{2}}) $ and therefore better than that of the normal one. Examples of studentized statistics that can appear in the analysis of autoregressive models are shown.

62E20, 62M10, 62G09