The prototype of this problem is about how do to estimate sampling errors in PCA analysis, and it seems harmless so we restate this in terms of SVD.
Consider $x_1,\dots,x_n,x_{n+1}\in \mathbb{R}^m$ sampled from multi-dimensional gaussian distribution $N(0,\Sigma)$. For our purpose we can assume $\Sigma$ is a diagonal matrix with diagonal term $\sigma_1^2,...,\sigma_m^2,\,\text{s.t.}\,,\,\sigma_1>...>\sigma_m>0$. Then do SVD for matrix $(x_1,\dots,x_n)=U*W*V^{T}$ and let the first $k$ row vectors of $U$ be $v_1,\dots,v_k$. (Which is the analog of principal components in PCA)
Then we project $x_{n+1}$ to the linear span of $v_1,\dots,v_k$, assume we get $x^{\prime}_{n+1}$. And I'm interested in the expectation of $\|x_{n+1}-x^{\prime}_{n+1}\|_2^2$, namely the residual. If there is no sampling error, we would have $v_1=(1,0,\dots,0)^{T},\,v_2=(0,1,0,\dots,0)^{T},\dots$. Thus the expectation would be $\sigma_{k+1}^2+\dots+\sigma_{m}^2$. You may assume $m \gg k$ and $\sigma_1 \gg ... \gg \sigma_m$ if nessesary. I'm most interested in asymptotic behaviour of $n\rightarrow \infty$. The final answer might be $\sigma_{k+1}^2+\dots+\sigma_{m}^2+f(\sigma_1^2,...,\sigma_m^2)/n+O(1/n^2)$. Any hints for getting $f(\sigma_1^2,...,\sigma_m^2)$?