We address the problem of estimating the autocovariance matrix of a stationary process. Under short range dependence assumptions, convergence rates are established for a gradually tapered version of the sample autocovariance matrix and for its inverse. The proposed estimator is formed by leaving … the main diagonals of the sample autocovariance matrix intact while gradually down-weighting oï¿½-diagonal entries towards zero. In addition we show the same convergence rates hold for a positive deï¿½nite version of the estimator, and we introduce a new approach for selecting the banding parameter. The new matrix estimator is shown to perform well theoretically and in simulation studies. As an application we introduce a new resampling scheme for stationary processes termed the linear process bootstrap (LPB). The LPB is shown to be asymptotically valid for the sample mean and related statistics. The eï¿½ectiveness of the proposed methods are demonstrated in a simulation study.