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](https://www3.stat.sinica.edu.tw/statistica/oldpdf/A17n418.pdf)) 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](https://mathoverflow.net/q/315221/159685) or [this](https://math.stackexchange.com/q/444700/498394)), and by the law of large number it has a clearly well-defined limit.



Convergent condition of the high-dimensional submatrix of some orthogonal matrix
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 $p/r\gtrsim 5$.


![](https://i.stack.imgur.com/Ihzk4.png)
![](https://i.stack.imgur.com/0rkrA.png)
Blue points are $(p-r+1)/p$, gold points are the number $Z$ for a single randomly chosen $V$.


