Asymptotic behavior of the covariance-scale CRB

Determine the asymptotic behavior, as the sample size tends to infinity, of the Cramér–Rao bound for the covariance-scale parameter $\sigma_x^2$ of the Student’s-$t$ stationary AR(1) compound Gaussian process and, more generally, of the covariance-scale parameter when $a_{1,n}+n a_{2,n}$ converges to zero at a distribution-dependent rate.

Background

For the scalar nonzero-mean AR(1) process, the paper derives an exact finite-sample expression for ${\rm CRB}_{\mathbf y_n}(\sigma_x^2)$ involving the coefficient combination $a_{1,n}+n a_{2,n}$. For purely continuous texture distributions, or for nondegenerate unknown-density-generator settings, the limiting Fisher information alone yields $a_{1,n}+n a_{2,n}\to 0$.

The paper therefore states that the asymptotic behavior of the covariance-scale CRB cannot be inferred from the limiting Fisher information and must instead be analyzed separately for each texture distribution and convergence rate. A Student’s-$t$ example is then treated explicitly, but no general characterization is provided.