Uniform convergence of autocovariances

We establish convergence rates for sample autocovariances and autocorrelations of stationary time series which have a moment of order αα, 2<α<42<α<4. Convergence is uniform over all lags ℓℓ where 0<ℓ⩽hN0<ℓ⩽hN where hNhN is increasing with sample size N at a prescribed rate. Best strong convergence rates are obtained which complement the results of An et al. [1982. Autocorrelation, autoregression and autoregressive approximation. Ann. Statist. 10, 926–936], where fourth moments are assumed.