As is known, the covariance matrix of a set of random vectors $\{\mathbf{x}_i\}_{i=1}^N$ can be estimated by their sample covariance matrix: $\mathbf{\hat R}:=\frac{1}{N}\sum_{n=1}^N\mathbf{x}_n\mathbf{x}_n^T$
In popular subspace methods such as PCA and MUSIC, the eigenvectors of $\mathbf{\hat R}$ is used to estimate the signal subspace and noise subspace.
As far as I know, there are mainly two streams of results about the relation between the sampled eigenvectors $\mathbf{\hat U}:=[\mathbf{\hat u}_1,\mathbf{\hat u}_2,\cdots]$ of $\mathbf{\hat R}$ and the true eigenvectors $\mathbf{U}:=[\mathbf{u}_1,\mathbf{u}_2,\cdots]$ of $\mathbf{R}$,
Firstly, under Gaussian assumption, the sampled eigenvectors $\mathbf{\hat U}$ obey an asymptotic normal distribution (in the sense of large samples N)
$$\mathbf{\hat U}\simeq \mathcal{N}(\mathbf{U},\mathbf{\Sigma})$$
these results were conducted by T.W. Anderson, P. Stoica, etc.
And secondly, the sampled eigenvectors $\mathbf{\hat U}$ can be modeled as a perturbation version of the true eigenvectors $\mathbf{U}$, that is $$\mathbf{\hat U}=\mathbf{U} + \delta\mathbf{U}$$
where the perturbation $\delta\mathbf{U}$ is related with the perturbation caused by finite samples, these results were conducted by G Stewart, F Li, etc.
My question is, how should we evaluate these two kinds of analysis, I mean, which one is better from what aspect, and why?
And, are there new and more advanced results about the relationship between the $\mathbf{\hat U}$ and $\mathbf{U}$ mentioned above? Is there any general theoretical background lying behind these analyzes?