In the context of AR(1) model, the following $n \times n$ matrix plays an important role:
$$ V(\rho) = \{\rho^{|i - j|}\}_{1 \leq i, j \leq n} \ \ \ (\rho \in (0, 1)) . $$
I am interested in asymptotic properties of the following:
$$ \hat I_n := V(\rho)^{1/2} V(\hat\rho)^{-1} V(\rho)^{1/2} $$
where $\hat \rho$ is an estimator of $\rho.$ Intuitively, this matrix is close to the $n \times n$ identity matrix, but the problem is that the size $n$ grows, so we cannot simply write like $\hat I_n \to I_n.$ Still, I believe that $\hat I_n$ is close to the identity matrix in a sense; for instance, all the eigenvalues go to one.
Let $0 \leq \lambda^{(n)}_1 \leq \cdots \leq \lambda^{(n)}_n$ be the eigenvalues of $\hat I_n.$ My conjectures are
(1) for fixed $i,$ $\lambda^{(n)}_i \overset{p}{\to}1;$
(2) $\lambda^{(n)}_n \overset{p}{\to} 1$ and $\lambda^{(n)}_1 \overset{p}{\to} 1;$
(3) (hopefully) $\sqrt{n} (\lambda^{(n)}_n - 1) \overset{d}{\to} \text{some distribution}$ and $\sqrt{n} (\lambda^{(n)}_1 - 1) \overset{d}{\to} \text{some distribution}.$
Are these correct under some conditions? Thanks!