  Multivariate time series present many challenges, especially when they are
high dimensional. The paper's focus is twofold. First, we address the subject
of consistently estimating the autocovariance sequence; this is a sequence of
matrices that we conveniently stack into one huge matrix. We are then able to
show consistency of an estimator based on the so-called flat-top tapers; most
importantly, the consistency holds true even when the time series dimension is
allowed to increase with the sample size. Second, we revisit the linear process
bootstrap (LPB) procedure proposed by McMurry and Politis [J. Time Series Anal.
31 (2010) 471-482] for univariate time series. Based on the aforementioned
stacked autocovariance matrix estimator, we are able to define a version of the
LPB that is valid for multivariate time series. Under rather general
assumptions, we show that our multivariate linear process bootstrap (MLPB) has
asymptotic validity for the sample mean in two important cases: (a) when the
time series dimension is fixed and (b) when it is allowed to increase with
sample size. As an aside, in case (a) we show that the MLPB works also for
spectral density estimators which is a novel result even in the univariate
case. We conclude with a simulation study that demonstrates the superiority of
the MLPB in some important cases.
