Convergent condition of the high-dimensional submatrix of some orthogonal matrix

Question

Let $\mathbf{V}$ be a $p\times p$ orthogonal matrix (i.e., $\mathbf{V}\mathbf{V}^\top = \mathbf{V}^\top \mathbf{V} = \mathbf{I}$) whose columns are $$ \mathbf{V} = \begin{bmatrix} \mathbf{v}_1 & \mathbf{v}_2 & \cdots & \mathbf{v}_p \end{bmatrix}. $$

Denote its submatrix as $$ \mathbf{V}_r = \begin{bmatrix} \mathbf{v}_r & \cdots & \mathbf{v}_p \end{bmatrix}. $$ for some $r>0$, so the $\mathbf{V}_r$ is a $p\times (p-r+1)$ matrix.

Then the question is

Question.

Under what condition $p^{-1} \mathbf{1}^\top \mathbf{V}_r \mathbf{V}_r^\top \mathbf{1} \rightarrow C$ for some well-defined value $C$ as $p\rightarrow \infty$? (Here $r$ is fixed, and $\mathbf{1}$ is a vector of ones.)

I apologize that this question would be too naive, but I just want to find some intuition about under what special structure of $\mathbf{V}_r$ makes $p^{-1} \mathbf{1}^\top \mathbf{V}_r \mathbf{V}_r^\top \mathbf{1}$ converges. This question has a quite important meaning in the high-dimensional inference of some factor model because under the factor model $$ \mathbf{y}_t = \mathbf{B} \mathbf{f}_t + \mathbf{\varepsilon}_t, $$ where $\mathbf{f}_t$ is a $r$-dimensional random factors, and its covariance
$$ \mathbf{\Sigma}_y = \mathbf{B} \mathbf{\Sigma}_f \mathbf{B}^\top + \mathbf{\Sigma}_{\varepsilon} = \mathbf{V} \mathbf{\Lambda} \mathbf{V}^\top, \qquad (\text{by the singular value decomposotion}.) $$ its submatrix of eigenvectors $\mathbf{V}_r$ corresponds to the idiosyncratic part $\mathbf{\Sigma}_{\varepsilon}$. So $p^{-1} \mathbf{1}^\top \mathbf{V}_r \mathbf{V}_r^\top \mathbf{1}$ corresponds to the (normalized) variance of equally weighted portfolio due to the idiosyncratic risk. To estimate this value, we have to guarantee that it has a well-defined limit. So it would be very appreciated if you let me know if there is any special structure for that (or is there any good material to study).

Here what I have done. Notice that if $r=1$, then it is just $$ p^{-1} \mathbf{1}^\top \mathbf{V}_r \mathbf{V}_r^\top \mathbf{1} = p^{-1} \mathbf{1}^\top \mathbf{V} \mathbf{V}^\top \mathbf{1} = 1, $$ so it is meaningful to consider $r>1$. It is easy to show that $p^{-1} \mathbf{1}^\top \mathbf{V}_r \mathbf{V}_r^\top \mathbf{1}$ is bounded: $$ 0 \le p^{-1} \mathbf{1}^\top \mathbf{V}_r \mathbf{V}_r^\top \mathbf{1} \le p^{-1} \| \mathbf{1}\|^2 \| \mathbf{V}_r\|^2 = 1, $$ where $\| \mathbf{V}_r\|$ is a spectral norm of the matrix $\mathbf{V}_r$. So it has a convergent subsequence, and the problem is that showing every subsequence converges to the same limit $C$. Also, notice that this value is invariant under any rotation $\mathbf{O}$ because $$ \mathbf{1}^\top \mathbf{V}_r \mathbf{O} \mathbf{O}^\top \mathbf{V}_r^\top \mathbf{1} = \mathbf{1}^\top \mathbf{V}_r \mathbf{V}_r^\top \mathbf{1}, $$ so this value is the same for any $\mathbf{V}_r'$ who has the same span of $\mathbf{V}_r$. As an illustrative example of $\mathbf{V}_r$, $\mathbf{v}_i = \mathbf{e}_i$ have a clearly well-defined limit of $p^{-1} \mathbf{1}^\top \mathbf{V}_r \mathbf{V}_r^\top \mathbf{1} = (p-r+1)/p \rightarrow 1$. Also, Paul (2007, This paper) shows that eigenvector of covariance matrix of i.i.d. Gaussian normal $\mathcal{N}(\mathbf{0}, \mathbf{I})$ distributed uniformly on the surface of unit sphere $\mathbb{S}^{p-1}$. This implies that each element of the eigenvector distributed standard normal (see this or this), and by the law of large number it has a clearly well-defined limit.

Answer 1

For a given matrix $V$ this number $Z=p^{-1} \mathbf{1}^\top \mathbf{V}_r \mathbf{V}_r^\top \mathbf{1}$ will converge to $1$ if you take the limit $p\rightarrow\infty$ at fixed $r$. If you average over $V$ (with the Haar measure), the convergence to $(p-r+1)/p$ will apply also for finite ratio $p/r$.

At the left you see a plot of $Z$ for a single matrix $V$ (drawn from $O(p)$ with the Haar measure), as a function of $p$ for $r=5$. Notice the large statistical fluctuations, which are only damped out when $p/r\rightarrow\infty$. At the right the data has been averaged over 10 random choices of $V$, and the curve $(p-r+1)/p$ is followed more closely already for $p/r\gtrsim 5$.

Blue points are $(p-r+1)/p$, gold points are the number $Z$ for a single randomly chosen $V$.