Circulant approximations of the inverses of Toeplitz matrices and related quantities with applications to stationary random processes

The circulant approximation of vector and quadratic forms involving the inverse of a Toeplitz covariance matrix R is addressed. First, a result is presented which increases the rate of convergence of the average matrix error under certain conditions on
\undertilde{r}
, the vector which defines R. Concerning vector and quadratic operations using R
^-1
, it is noted that if
\undertilde{x}
is AR(p), then the p-banded, near-Toeplitz structure of R
^-1
results in an O(1/N)-type mean convergence of associated errors.