  The class of complex random vectors whose covariance matrix is linearly
parameterized by a basis of Hermitian Toeplitz (HT) matrices is considered, and
the maximum compression ratios that preserve all second-order information are
derived --- the statistics of the uncompressed vector must be recoverable from
a set of linearly compressed observations. This kind of vectors arises
naturally when sampling wide-sense stationary random processes and features a
number of applications in signal and array processing.
  Explicit guidelines to design optimal and nearly optimal schemes operating
both in a periodic and non-periodic fashion are provided by considering two of
the most common linear compression schemes, which we classify as dense or
sparse. It is seen that the maximum compression ratios depend on the structure
of the HT subspace containing the covariance matrix of the uncompressed
observations. Compression patterns attaining these maximum ratios are found for
the case without structure as well as for the cases with circulant or banded
structure. Universal samplers are also proposed to compress unknown HT
subspaces.
