Approximations of the eigenvalues of the covariance matrix of a first-order autoregressive process

This paper provides explicit estimates of the eigenvalues of the covariance matrix of an autoregressive process of order one. Also explicit error bounds are established in closed form. Typically, such an error bound is given by $\text{ε}{}_{\mathrm{k}}\text{= (}\text{4}\text{(n+1)}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{ρ}{}^2\text{sin}\text{(}\text{kπ}\text{(n+1)}\text{)}$, so that the approximations improve as the size of the matrix increases. In other words, the accuracy of the approximations increases as direct computations become more costly.